Asymptotic behavior of scalar convection-diffusion equations
Enrique Zuazua
math.AP

	
	
	
	
	
	
	
	
	
	
	
	
	Asymptotic behavior of scalar convection-diffusion equations[Translated from the original Spanish version <cit.> and bibliographically updated by Borjan Geshkovski.]
    Enrique Zuazua 
 
 March 19, 2020
    19 March, 2020
=====================================================================================================================================================================

Acknowledgments:  These Notes originate in a course that I delivered at the Institute of Mathematics  of the Universidad Federal de Rio de Janeiro (UFRJ) in March 1991. I had the honor of visiting that institution under the invitation and hospitality of Prof. Carlos Frederico Vasconcellos.
		At that time the Notes were published in Spanish with the title "Comportamiento   asintótico   de   ecuaciones   escalares   de
convección-difusión" in 1993 in the Lecture Notes Series of the Institute. 

Recently, our PhD student Borjan Geshkovski, who has worked in this area, among others, was kind enough to carefully translate the Notes to English, updating the bibliography and catching up some typos.

During the preparation of this last version we received funding from the Alexander von Humboldt-Professorship program, the European Research Council (ERC) Advanced Grant DyCon, the European Unions Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No.765579-ConFlex, the Air Force Office of Scientific Research under Award NO: FA9550-18-1-0242, the Grant MTM2017-92996-C2-1-R COSNET of MINECO (Spain), the Grant ICON-ANR-16-ACHN-0014 of the French ANR and the  Transregio 154 Project "Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks" of the German DFG.
Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks.
	
			
			
			
			
	
	

		
		

CHAPTER: INTRODUCTION

		
	In these notes, we address the problem of asymptotic behaviour when time goes to infinity of the solutions to scalar convection-diffusion equations of the form
		
    u_t-Δ u    = a·∇(F(u))  in ^N ×(0,∞) 
     
    			u(x,0)   =u_0(x)

		where a is a constant vector in ^N, F∈ C^1() is such that
		F(0)=0 and u_0∈ L^1(^N). Henceforth · denotes the scalar product
		in ^N.
		The case a=0 corresponds to the linear heat equation.
		
	In spatial dimension N=1 and when
		
    F(u)=1/2 u^2

		and a=1, equation (<ref>) was first introduced by H. Bateman <cit.> and J. M. Burgers <cit.> as an approximation for equations of fluid-flow, and has since been known as the Burgers equation. 
It is a very simple model which combines nonlinear wave propagation with the effect of heat conduction.
		
These equations are also a model for the displacement of a fluid in a porous medium, when the effect of capilarity is taken into account (cf. S. E. Buckley and M. C. Leverett <cit.>; see also D. W. Peaceman <cit.> for a numerical treatment).

More generally, (<ref>) is a models the regularizing effect of a (small) viscosity in a hyperbolic conservation law.
		
The asymptotic behaviour of the solutions to (<ref>)–(<ref>) is strongly linked with the behavior of the initial data u_0  when  |x|→∞. 
In order to show this fact, we consider the case when the spatial dimension N equals one: N=1.
Let a=-1, let F be a convex function and 
    u_± =lim_|x|→±∞ u_0(x).


Then:

		
			
  (a)
			If u_-<u_+, the solutions to (<ref>)–(<ref>),  as t→∞,
			converge in L_ loc^p(), 1≤ p<∞, to a rarefaction wave p(x/t) which is an entropy solution of the Riemann problem with data u_- and u_+ for the hyperbolic conservation law: 
	
    u_t+(F(u))_x=0,

			(cf. A. M. Il'in and O. A. Oleinik <cit.>, <cit.> and E. Harabetian
			<cit.>). This type of results have been further extended by Z. P. Xin <cit.> for
			 2× 2 systems in dimension N=1, and by Z. P. Xin <cit.> y
			E. Harabetian <cit.> for scalar equations in multiple spatial dimensions dimensions (for the case where when F is not convex, see H. F. Weinberger
			<cit.>).
			
			
  (b)
			If u_+>u_-, A. M. Il'in and O. A. Oleinik <cit.> proved the existence and stability of solutions of  “travelling waves” type: u=u(x-ct). This type of results has been further extended to systems in multiple spatial dimensions by J. Goodman <cit.> and T. P. Liu <cit.> and scalar equations in multiple spatial dimensions by J. Goodman
			<cit.>.
			
			
  (c)
			if u_0∈ L^1() (which corresponds to the case u_-=u_+=0) and if
			
    F(u)=1/2 u^2

			(Burgers equation), thanks to the Hopf-Cole transformation, it can be shown that (<ref>) admits a one-parameter family of self-similar solutions
			
    u_M(x,t)=t^-1/2f_M(x/√(t))

			where the profiles f_M satisfy
			
    ∫^∞_-∞f_M(x) dx=M.

		
		
		On another hand,  it can be shown that if u_0∈ L^1() and
		
    ∫^∞_-∞u_0(x) dx=M,
 the solution to (<ref>)–(<ref>)
		behaves like u_M  when  t→∞. therefore, in this case, the asymptotic behaviour of the system is described by a one-parameter family of self-similar solutions {u_M}_M∈. The parameter M corresponds to the mass of the solution, a quantity which is conserved along the entire trajectory:
		
    ∫^∞_-∞u(x,t) dx=∫^∞_-∞u_0(x) dx,
    		    ∀ t>0.

		This type of results have been extended by S. Kawashine <cit.>,
		T. P. Liu <cit.> and I. L. Chern and T. P. Liu <cit.> for systems of viscous conservation laws in dimension N=1.
		The extension to scalar equations in dimensions N>1 was done by M. Escobedo and E. Zuazua <cit.>, <cit.>.
		
		In these notes we address exclusively the case of initial data
		u_0∈ L^1(^N).
		To fix ideas, we first consider the case of a particular homogeneous nonlinearity such as
		
    u_t-Δ u=a·∇(|u|^q-1u)    en ^N×(0,∞)
    
    		 u(x,0)=u_0(x)    en ^N.

		Integrating the equation (<ref>) over ^N with respect to the spatial variable
		x it can be observed that
		
    d/dt∫_^N u(x,t) dx=0.

		Thus, the mass of the solution is conserved over time.
		
		The question at hand is the study of how the mass is
distributed in space when  t→∞.
		
		As we will show later, we have the following estimate
of the decay of solutions to (<ref>) in L^p(^N):
		
    u(t)_p≤ C_pt^-,    ∀ t>0

		for all p∈ [1,∞] with C_p>0 depending on u_0_1.
		Henceforth ·_p denotes the usual norm in L^p(^N).
		
		The problem at hand, formulated more precisely is the
behavior of t^ u(t) in L^p(^N)  when  t→∞.
		
		When a=0, (<ref>) is the linear heat equation and its solutions behave like u_M=MG, where M is the mass of the solution
		
    M=∫_^Nu_0(x) dx
 and G is the heat kernel, i.e., the fundamental solution of the linear heat equation:
		
    G(x,t)=(4π t)^-N/2exp(-|x|^2/4t).

		That is, for all p∈[1,∞] it holds
		
    t^u(t)-u_M(t)_p→ 0     when     t→∞

		It is again observed that asymptotic behavior is governed by a one-parameter family of self-similar solutions: u_M=MG.
		
		The first question that arises in the general case (<ref>) is the existence of self-similar solutions.
		
		The equation (<ref>) is invariant with respect to the scaling
		
    u_λ (x,t)=λ^1/(q-1) u(λ x,λ^2t),
    		    λ>0.

		The self-similar solutions of (<ref>)  are the ones that remain invariant with respect to this scaling, i.e.,
		
    u_λ (x,t)=u(x,t),    ∀λ>0, ∀(x,t)∈^N ×(0,∞).

		It is easy to check that u satisfies (<ref>) if and only if it is of the form
		
    u(x,t)=t^-1/[2(q-1)] f(x/√(t)).

In view of the conservation of mass (<ref>), a function u that admits the expression (<ref>) with profile f∈ L^1(^N) can only be a solution of (<ref>) if q=1/N+1.
		
		We therefore see that the system (<ref>) can only admit self-similar solutions in L^1(^N) for the exponent q=1+1/N. In dimension N=1 it holds q=2 which corresponds to the Burgers equation and which, as we saw earlier, admits self-similar solutions.
		
		When q=1+1/N, the self-similar solutions are 
		of the form
		
    u(x,t)= t^-N/2 f(x/√(t)).

		The heat kernel admits this expression for the profile
		
    f(x)= (4π)^-N/2exp(-|x|^2/4).

		Finding a solution of the form (<ref>) to the equation (<ref>) with
		q=1+1/N is equivalent to solving the elliptic equation:
		
    -Δ f-x·∇ f/2-N/2 f =a·∇
    			(|f|^1/Nf)    in ^N.

		In <cit.>, using a fixed point method we prove that for any M∈ the equation (<ref>) admits a unique solution f_M such that
		
    ∫_^Nf_M(x) dx=M.

		This solution decays exponentially to zero  when  |x|→∞.
		
		Later, in <cit.> we prove that the general solution to (<ref>) with
		q=1+1/N behaves like the self-similar solution 
		
    u_M(x,t)=t^-N/2f_M(x√(t))
 if		
	
    ∫_^Nu_0(x) dx=M.
 
	That is, we prove that  (<ref>) is verified. This is an extension of the results mentioned above for the Burgers equation in dimension N=1 and for the linear heat equation.
		
		When q>1+1/N, as we proved in <cit.>, the
		solutions to (<ref>) behave like the heat kernel, that is, (<ref>) holds with u_M=MG. In this case, we say that the system has
		weakly non-linear behavior.
		
		Let us consider next the equation with linear convection
		
    u_t-Δ u=a·∇ u    en ^N×(0,∞),
    
    		 u(0)=u_0

		which corresponds to q=1.
		
		The function
		
    v(x,t)= u(x-at,t)

		satisfies
		
    v_t-Δ v=0, 
    
    		v(0)=u_0.

		
		In the case of the linear heat equation, v behaves like u_M=MG, where 
    M=∫_^Nu_0(x) dx.
 Thus,  when 
		t→∞, u behaves like MG(x+at,t), that is,
		(<ref>) holds for u_M(x,t)=MG(x+at,t).
		
		The asymptotic behavior of the system  when  1<q<1+1/N is of different nature. To describe it, it is appropriate to distinguish the case N=1
		from the case N≥ 2. We begin with the one-dimensional case N=1 which was solved by M. Escobedo, J. L. Vázquez and E. Zuazua in <cit.>.
		
		In <cit.> it is proven that if N=1 and 1<q<2 the solution to (<ref>) satisfies 
		
    t^1/q(
    					1-1/p)u(t)-u_M(t)_p→ 0     when     t→∞

		
		for all 1≤ p<∞ where u_M=u_M(x,t) is the unique entropic solution of the hyperbolic system
		
    u_t=a(|u|^q-1u)_x   en ×(0,∞)
    
    		u(0)=Mδ.

		The uniqueness of the entropic solution to (<ref>) was shown T. P. Liu and
		M. Pierre <cit.>. The entropic solution of (<ref>) is self-similar and for a=-1/q is of the form
		
    u_M(x,t)=(x/t)^1/(q-1)
    				X_(0,r(t))

		X_(0,r(t)) being the characteristic function of the interval (0,r(t)) and
		
    r(t)=cM^(q-1)/q t^1/q,    
    			c=(q/q-1)^(q-1)/q.

		In this case we see therefore that the effect of the term of diffusion of the equation (<ref>) disappears when  t→∞. We are in the presence of a		strongly non-linear behavior.
		
		When N≥ 2 and 1<q<1+1/N there is an analogue result but this time the diffusion disappears only in the direction of the convection a. For example, if a=(1,0,…,0) the asymptotic behavior it is given by the self-similar entropy solutions of the hyperbolic-parabolic equation
		
    u_t-Δ'u=∂_x_1(|u|^q-1u|)   en ^N
    				×(0,∞),
    
    			u(0)=Mδ

		Δ' being the laplacian in the variables
		x_2,…,x_N (see <cit.>).
		
		Let's mention that, in this case 1<q<1+1/N, One of the main difficulties is to test the following estimate on the rate of decay
in L^∞(^N) of the solutions to (<ref>):
		
    u(t)_∞≤ Ct^-(N+1)/2q

		which, since 1<q<1+1/N, is faster than in other cases, both in the linear and in the self-similar or in the weakly non-linear.	
		In the general case, the system (<ref>) presents these four types of asymptotic behaviors (self-similar, weakly non-linear, linear and strongly non-linear convection) depending on the behavior of nonlinearity F at the origin.
			
		As we have seen, in the case of linear convection, through a simple change of variables, the equation is reduced to the linear heat equation. Therefore, we will deal primarily with the study of behavior
asymptotic self-similar and weakly non-linear. In Chapter 7 we will describe
briefly the results obtained in <cit.>.
		
		In the case of weakly non-linear asymptotic behavior, after obtaining optimal estimates of the decay of the L^p(^N) norms of the solutions, the result is easily obtained by working on the integral equation associated with the convection-diffusion equation.
In the self-similar case we will adopt the point of view introduced in <cit.>. First, the self-similar variables are introduced since, in these, the profiles of the self-similar solutions are the stationary solutions of the new parabolic equation and the self-similar behavior of the general solution is equivalent to the convergence of the new trajectories to the stationary solutions. Working within the functional framework of the Sobolev spaces with weight introduced by M. Escobedo and O. Kavian <cit.> , we will test the pre-compactness of the trajectories. Finally we will use a dynamic systems argument that allows us to simultaneously prove the existence of stationary solutions and the convergence of trajectories towards these stationary solutions. At this point, the strong contraction in L^1 (^N) which is verified by the semigroup associated with  (<ref>), will be fundamental.
The notes are organized as follows. In Chapter 1 we will study the linear heat equation and in Chapter 2 the Burgers equation
in dimension N = 1 by means of the Hopf-Cole transformation. In Chapter
3 we will study the linear heat equation in the self-similar variables. In
Chapter 4 will address the Cauchy problem (<ref>)–(<ref>) and prove a
result of existence and uniqueness of solutions for data u_0 ∈ L^1(^N), as well
as estimates of solution decay in L^p(^N). In Chapter 5 we will prove weakly nonlinear behavior for nonlinearities F that behave like F(s)=|s|^q-1s
with q>1+1/N. 
Chapter 6 we will show the self-similar asymptotic behavior when q=1+1/N.
Finally, in Chapter 7, we will describe the main results obtained in <cit.> for the case
		q<1+1/N, we will mention some extensions of the results of these notes as well as some open problems.
	
	

 §.§ Subsequent works

	Hereinafter, we present a non exhaustive list of developments in the field of large time asymptotics for convection-diffusion equations since these notes were first written.
		
			
  * Subsequent studies on scalar convection-diffusion and Hamilton-Jacobi equations include <cit.>;
			
  * Large time asymptotics for simplified fluid-structure interaction models are studied in		<cit.>;
			
  * An exhaustive treatise of the large time asymptotics of scalar hyperbolic conservation laws is <cit.>, while viscous conservation laws are addressed in <cit.>;
			
  * The large time asymptotics for numerical schemes of such equations is studied in <cit.>;
			
  * The large time asymptotics for non-local diffusion-convection equations are studied in <cit.>;
			
  * The case of degenerate diffusion with or without convective terms is addressed in <cit.>	
			
  * For the large time asymptotics of the Navier-Stokes equations, we refer to <cit.>
			
  * Finally, for scaling properties and large time asymptotics for damped or dissipative wave equations we refer to  <cit.>.	
		

		
		

CHAPTER: THE LINEAR HEAT EQUATION

		
		

§ INTRODUCTION

		
		Let us consider the linear heat equation
		
    u_t-Δ u=0    en ^N ×(0,∞),
    
    		u(0)=u_0.

		As is well known, the fundamental solution or heat kernel (solution of  (<ref>) with u_0=δ_= Dirac mass at the origin) is given by the formula
		
    G(x,t)=(4π t)^-N/2exp(-|x|^2/4t).

	Given u_0∈ L^1(^N) the solution to (<ref>) is given by convolution with the heat kernel:
		
    u(x,t)   =[G(·,t)*u_0(·)](x)
       =(4π t)^-N/2∫_^Nexp(-|x-y|^2/4t)
    			u_0(y) dy.

		It is the unique solution of
		(<ref>) in the class u∈ C((0,∞);L^1(^N)). Thanks to
		regularizing effect of the heat kernel it holds u∈ C^∞(^N×(0,∞)).
		
		Integrating the equation (<ref>) in ^N we obtain
		
    d/dt∫ u(x,t) dx=0.

		(Henceforth  ∫ denotes the integral in all ^N.) From (<ref>) it follows that the mass of the solution is conserved over all time.
		
		The question that interests us is to study the way in which the mass of the
solution is distributed in space when t→∞.
		
		
		Calculating the L^p(^N) norm of the heat kernel we obtain
		
    G(t)_p≤ C_pt^-,    ∀ t>0

		for all p∈[1,∞].
		Therefore, thanks to  the Young inequality we obtain
		
    u(t)_p=G(t)*u_0_p   ≤G(t)_pu_0_1 
       ≤ C_pt^-u_0_1,    ∀ t>0

		for all p∈[1,∞].
		
		The question at hand can be formulated a little more precisely: we wish to study the behavior of t^ u(t) in
		L^p(^N)  when  t→∞.
		
		As we mentioned in the introduction and as we will prove in the
next section, asymptotic behavior is given by u_M=MG, M being the mass of the solution, i.e.,
		
    M=∫ u_0(x) dx.

		This result can be seen by a scaling argument.
		
		Let u be the solution to (<ref>) with initial data u_0∈ L^1(^N) of mass
		M. It is easy to check that the rescaled function
		
    u_λ(x,t)=λ^Nu(λ x,λ^2t)

		is a solution to
		
    u_λ ,t- Δ u_λ =0         in
    			^N×(0,∞),
    
    		u_λ (x,0)=u_0,λ(x)=λ^Nu_0(λ x).

		
		On the other hand
		
    u_0,λ→ Mδ    when    λ→∞

		en the following sentido
		
    ∫ u_0,λ(x)φ(x) dx→ Mφ(0),
    		    when    λ→∞, ∀φ∈(^N).

		(Henceforth (^N) designates the space continuous and bounded functions in ^N.)
		
		Therefore, it is natural  natural to think that u_λ(t) converges to MG(t)
		 when  λ→∞ for all t>0 as u_M(x,t)=MG(x,t) is the solution to the heat equation with with  initial data Mδ.
		
	If this convergence takes place in L^1(^N) for t=1 it holds
		
    u_λ(·,1)→ MG(·,1)    in  L^1(^N),     when  λ→∞

		and this is equivalent to
		
    u(t)-MG(t)_1→ 0     when   t→∞.

		n the next section we will demonstrate this result rigorously. We will see that
		
    u_λ(·,1)→ MG(·,1)    in  L^p(^N),     when  
    		t→∞

		for all 1≤ p≤∞.
		This answers the question we had
raised.
		
		
		
		

§ ASYMPTOTIC BEHAVIOR

		
		The following result holds. 
				
		
			Let u_0∈ L^1(^N) be such that M=∫ u_0(x) dx.
			Then, the solution u=G(t)*u_0 de
			(<ref>) satisfies
			
    t^u(t)-MG(t)_p→ 0     when    t→∞

			for all p∈[1,∞].
		
		
		The proof of this theorem is based on the following lemma from
		<cit.>.
		
		
			(a) 
			For all p∈[1,∞] there exists a constant C_p>0 such that
			
    G(t)*φ_p ≤
    				C_pt^--1/2φ_L^1(^N;|x|),
    				    ∀ t>0

			for all φ∈ L^1(^N;1+|x|) such that
			∫φ(x) dx=0. 

			
			
			(b) For all p∈[1,∞] there exists a constant
			C'_p>0 such that if φ∈ L^1(^N;1+|x|) satisfies
			∫φ(x) dx=M
			then it holds
			
    G(t)*φ -MG(t)_p≤
    				C'_pt^--1/2φ_L^1(^N;|x|),
    				    ∀ t>0.

		
		
		
			We used the space L^1 with weights:
			
    L^1(^N;1+|x|)={ φ∈ L^1(^N): ∫ |φ(x)|(1+|x|) dx
    			<∞ }.

			The ·_L^1(^N;|x|) norm is, by definition,
			φ_L^1(^N;|x|)=∫|φ(x)||x| dx.
		
		
		 

			(a) It holds
				
    (G(t)*φ)(x)=(4π t)^-N/2∫exp(
    				-|x-y|^2/4t)φ(y) dy

				and since ∫φ(x) dx=0, we obtain
				
    (G(t)*φ)(x)=(4π t)^-N/2∫[exp(-|x-y|^2/4t)
    				-exp(-|x|^2/4t)]φ(y) dy.

				By Taylor's theorem we know that 
								
    exp(-|x-y|^2/4t)
    				-exp(-|x|^2/4t)=1/2t∫_0^1 y·(x-θ y)
    				exp[-(x-θ y)^2/4t] dθ

				and therefore
				
    (G(t)*φ)(x) 
       =(4π  t)^-N/2/√(t)∫_0^1∫y·(x-θ y)/2√(t)exp(-|x-θ
    							y|^2/4t)φ(y) dy dθ.

				Taking L^∞ norms it holds:
				
    G(t)*φ_∞
       ≤ (4π)^-N/2
    					t^-N/2-1/2φ_L^1(^N;|x|)sup_x,y∈^N∫_0^1{|x-θ y|/2√(t)exp(-|x-θ y|^2/4t) dθ}
       ≤(4π)^-N/2 |z|exp (-|z|^2)_∞ t^-N/2-1/2φ_L^1(^N; |x|).

				Therefore we obtain (<ref>) for p=∞ with C_∞=(4π)^-N/2
				|z| exp (-|z|^2) _∞.

				Taking L^1 norms in (<ref>) and using the Fubini theorem we obtain:
				
    G(t)*φ_∞
       ≤(4π)^-N/2 t^-N/2-1/2∫_0^1∫ 2|x-θ y|/2√(t)exp(-|x-θ y|^2/4t)
    						|yφ(y)| dx dy dθ
       ≤π^-N/2t^-1/2 |z|exp(-|z|^2)_1 ∫_0^1∫ |y|
    					|φ(y)| dy dθ
       ≤π^-N/2 |z|exp(-|z|^2)_1  φ_L^1(^N;
    						|x|)t^-1/2,     ∀ t>0.

				therefore, we obtain (<ref>) for p=1 with C_1=π^-N/2 |z|exp(-|z|^2)_1.
				
				The general case is obtained by interpolation:
				
    G(t)*φ_p ≤G(t)*φ_∞^(p-1)/pG(t)*φ_1^1/p
       ≤ 2^-N(p-1)/pπ^-N/2 |z|exp(-|z|^2)_∞^(p-1)/p |z|exp(-|z|^2)_1^1/pφ_L^1(^N;|x|)
    					t^--1/2.

				That is to say, we obtain (<ref>) with
				
    C_p=2^-N(p-1)/pπ^-N/2 |z|exp(-|z|^2)_∞^(p-1)/p |z|exp(-|z|^2)_1^1/p.

				This concludes the proof of part (a).
				
				(b) The proof of this part is analogous to the previous one. Indeed, since ∫φ(x) dx=M,
				
    (G(t) * φ) (x) -MG(x,t)
       =(4π t)^-N/2[∫exp(-|x-y|^2/4t) φ (y)  dy
    					-Mexp(-|x|^2/4t)]
       =(4π t)^-N/2∫[exp(-|x-y|^2/4t)
    					-exp(-|x|^2/4t)]φ(y) dy

				and therefore we can just redo the estimates from part (a).
		
		
		
			Lemma 1.2 can be proven easily using results from
			J. Duoandikoetxea and E. Zuazua <cit.> which ensures that for all
			u_0∈ L^1 (^N; 1+|x|) there exists f∈(L^1(^N))^N such that
			
    u_0=Mδ+÷ f

			with M=∫ u_0 dx. The vector-valued function f can be computed explicitly. It holds
			
    f_j(x)=∫_0^1 x_j u_0(tx)t^N-1 dt.

			From this expression one deduces that  f_(L^1(^N))^N≤
			Cu_0_L^1(^N;|x|).
		
		
		
			
			Lemma 1.2 provides (<ref>) in the case when u_0∈ L^1(^N; 1+|x|). We obtain the general case by density. Given arbitrary u_0∈ L^1(^N) with ∫ u_0(x) dx=M we construct a sequence φ_n∈
			L^1(^N;1+|x|) such that ∫φ_n(x) dx=M For all
			n∈ y
			
    φ_n→ u_0    in  L^1(^N)   when   n→∞.

			We have
			
    t^G(t) * u_0 - MG(t)_p
       ≤ t^{G(t) *
    				φ_n - MG(t)_p + G(t) * (φ_n -u_0)_p}.

			By virtue of (<ref>) we have
			
    t^G(t) * (φ_n -u_0)_p≤ C_pφ_n -u_0_1

			and therefore, thanks to  (<ref>), given arbitrary ϵ>0, there exists
			k∈ such that 
			
    sup_t>0{t^G(t) * (φ_k -u_0)_p}≤ϵ/2.

			On the other hand, thanks to Lemma 1.2, for fixed k,
			
    t^G(t) * φ_k -MG(t)_p→ 0     when   t→∞

			and therefore, for t_0>0 sufficiently large it holds
			
    t^G(t) * φ_k -MG(t)_p≤ϵ,     ∀ t≥ t_0

			which implies (<ref>).
		
		
		

§ THE HEAT EQUATION WITH LINEAR CONVECTION

		
		Let a be a constant vector of ^N, and consider the following linear convection-diffusion equation:
		
    u_t-Δ u = a·∇ u    in ^N ×(0,∞) 
    
    			u(0)= u_0∈ L^1(^N).

		It can easily be shown that if u satisfies (<ref>) then
		
    v(x,t)=u (x-at,t)

		satisfies
		
    v_t-Δ v =0     in ^N×(o,∞)
    
    				v(0)=u_0.

		As the unique solution to (<ref>) in C([0,∞); L^1(^N))
		is given by
		
    v(x,t)=[G(t)*u_0] (x)

		undoing the change of variables (<ref>) we obtain that the unique solution
		of (<ref>) in C([0,∞); L^1(^N)) is the following:
		
    u(x,t)=[G(t)*u_0] (x+at).

		From this expression for u we deduce  that the solutions to (<ref>) have the same decay properties in L^p as the solutions to the
		linear heat equation, i.e., the solutions to (<ref>) satisfy (<ref>).
		
		From Theorem 1.1 we obtain the following result which provide
		the asymptotic behavior of the solutions to (<ref>).
		
		
			For all u_0∈ L^1(^N) such that ∫ u_0(x) dx=M la
			solution u to (<ref>) satisfies
			
    t^u(x,t)-MG(x+at,t)_p→ 0,     when   t→∞

			for all p∈[1,∞].
		
		
		

CHAPTER: THE BURGERS EQUATION

		
		

§ THE HOPF-COLE TRANSFORM

		
		Let us consider the viscous Burgers equation:
		
    u_t-u_xx= (u^2)_x    en ×(0,∞)
    
    			u(x,0)=u_0(x)   en .

		At the beginning of the 1950s, E. Hopf <cit.> and J. D. Cole <cit.>
		independently discovered that the equation (<ref>) can be
reduced to the linear heat equation by means of a simple change of
variables This change of variables is known since the Hopf-Cole transform.
		
		Given u solution to (<ref>) we define 
		
    v(x,t)=∫_-∞^x u(z,t) dz.

		
		It can easily be shown that  v satisfies
		
    v_t-v_xx=(v_x)^2    en ×(0,∞)
    
    			v(x,0)=v_0(x)=∫_-∞^x u_0(z) dz

		We define w(x,t)=exp(v(x,t)). So, w is a solution to
		the equation heat equation:
		
    w_t-w_xx= 0    en ×(0,∞)
    
    			w(x,0)=w_0(x)=exp(∫_-∞^x u_0(z) dz)

		We see therefore that if u is solution to (<ref>), so
		
    w(x,t)=exp(∫_-∞^x u(z,t) dz)

		satisfies (<ref>).
		
		On the other hand we see that if u_0∈ L^1() with
		∫_-∞^∞ u_0(x) dx=M and u_0_1=L
		then
		
			
			
    w_0∈ C()∩ L^∞() 
       e^-|L|≤ w_0(x)≤ e^|L|,    ∀ x∈
       lim_x→-∞ w_0(x)=1 
       lim_x→+∞ w_0(x)=e^M

		
		The solution w to (<ref>) is given by convolution with the one-dimensional heat kernel
		
    w(x,t)=[G(·,t)*w_0(·)](x)

		where
		
    G(x,t)=(4π t)^-1/2exp(-|x|^2/4t).

		
		Note that  w_0 does not belong to  L^1(). Therefore, the
		L^p estimates obtained in the previous chapter for the solutions
		of the linear heat equation with initial data in L^1 do not apply to w.
		
		As w_0∈ L^∞(), applying the Young inequality we obtain
		
    w(t)_∞≤G(t)_1 w_0_∞ = w_0_∞,
    			∀ t>0

		On the other hand, combining (<ref>)–(<ref>) with the explicit expression of
		w:
		
    w(x,t)=(4π t)^-1/2∫_-∞^∞exp(-|x-y|^2/4t)w_0(y) dy

		we easily obtain
		
		
			
			
    e^-|L|≤ w(x,t) ≤ e^|L|,   ∀ (x,t)∈×(0,∞) 
       lim_x→-∞ w(x,t)=1,  lim_x→+∞
    						w(x,t)=e^M,   ∀ t>0

		
		Inverting the transformation (<ref>) we obtain
		
    u(x,t)=(log(w(x,t)))_x=
    			w_x(x,t)/w(x,t).

		Differentiating the equation (<ref>) with respect to x we see that ω=w_x is
		solution to
		
    ω_t-ω_xx=0     in ×(0,∞)
    ω(x,0)=ω_0(x)=w_0(x) u_0(x).

		As w_0∈ L^∞() and u_0∈ L^1() We see that ω_0∈
		L^1(). Thus, the L^p estimates obtained in Chapter 1 for the solutions to the heat equation with initial data in L^1() can be applied to ω and we obtain
		
    ω(t)_p ≤ C_p t^-,     ∀ t>0

		for all p∈[1,∞].
		
		Combining (<ref>) and (<ref>) we deduce that the solution u=u(x,t) to (i)
		satisfies
		
    u(t)_p≤ C'_p t^-,    ∀ t>0

		for all p∈[1,∞].
		
		On the other hand, integrating the equation (<ref>) in  with respect to x one obtains
				
    d/dt∫_-∞^∞ u(x,t) dx=0.

		Thus the mass of the solution to (<ref>) is conserved with time.
		
		The Hopf-Cole transform allowed us to obtain an explicit form for the solutions to the Burgers equation, as well as L^p () estimates which hold that for the heat equation. 
		In the next sections we will see that this transformation also allows obtaining self-similar solutions to (<ref>), as well as the asymptotic behavior of the general solution.
		
		

§ SELF-SIMILAR SOLUTIONS

		
		The equation (<ref>) is invariant under the rescaling
		
    u_λ(x,t)=λ u(λ x,λ^2 t).

		
		Therefore, the problem arises of the existence of self-similar solutions that remain invariant, i.e., that satisfy
		
    u_λ=u,    ∀λ>0.

		It can easily be shown  that u satisfies (<ref>) if and only if 
		
    u(x,t)=t^-1/2 f(x√(t))

		with f(x)=u(x,1).
		
		Any solution to (<ref>) of the form (<ref>) with profile f∈ L^1() es
		solution the Cauchy problem:
		
    u_t-u_xx=(u^2)_x     in ×(0,∞)
    
    				u(x,0)= Mδ

		with M=∫_-∞^∞ f(x) dx and δ being the mass Dirac at the origin.
				
		For every M∈, the existence and uniqueness of the solution to (<ref>) can be expected, with the solution being self-similar.
		
		Therefore, for every M∈ we look for f_M∈ L^1() with
		∫_-∞^∞ f(x) dx=M and such that the function
		u_M(x,t)=t^-1/2 f_M(x/√(t)) satisfies (<ref>).
		
		It can easily be shown that  u_M=t^-1/2 f_M(x/√(t)) satisfies (<ref>) if and only if	
		
    -f”_M-xf'_M/2-1/2 f_M=(f^2_M)'   en 
    ∫_-∞^∞ f_M=M.

		This differential equation is easily solved using the Hopf-Cole transform
		We define 
		
    g_M(x)=exp(∫_-∞^x f_M(s) ds).

		Then (<ref>) transforms to
		
    -g”_M-xg'_M/2=0     in 
    lim_x→-∞ g_M(x)=1,   lim_x→-∞ g(x)=e^M.

		If we write
		
    g_M(x)=(e^M-1)∫_-∞^x ∫_-∞^x h(s) ds+1

		we see that (<ref>) is equivalent to
		
    -h_xx-xh_x/2-1/2 h=0     in 
    ∫_-∞^∞ h dx=1.

		Hence, (<ref>) is the equation for the profile of the self-similar solution of the linear heat equation and thus :
		
    h(x)=(4π)^-1/2exp(-x^2/4).

		On the other hand, undoing the change (<ref>) we obtain the
		self-similar solution to (<ref>):
		
    f_M(x)=(log(g_M(x)))_x = g'_M(x)/g_M(x)
    		=(e^M-1)h(x)/(e^M-1) ∫_-∞^x h(s) ds+1.

		The self-similar solutions to (<ref>) are thus of the form:
		
    u_M(x,t)=t^-1/2(e^M-1)h(x/√(t))/(e^M-1)
    				∫_-∞^x/√(t) h(s) ds+1.
	
The main difference observed in the self-similar profiles of the Burgers equation with respect to the self-similar profiles of the heat equation is its asymmetry with respect to the origin. This is due to the convection term.
		
		When M increases, f_M(x) increases and  when  M goes to infinity f_M
		converges to 
		
    f_∞(x)=h(x)/∫_-∞^x h(s) ds.

		The function f_∞ satisfies: lim_x→∞ f_∞ (x)=0 y
		lim_x→-∞ f_∞(x)=∞.
		
		The function u_∞(x,t)=t^-1/2 f_∞(x/√(t)) is a self-similar solution of the Burgers equation with profile f_∞ which does not belong to L^1.
		
		

§ ASYMPTOTIC BEHAVIOR

		
		Given u_0∈ L^1(), we have seen that the Burgers equation (<ref>) admits a unique solution that we have explicitly obtained by means of the Hopf-Cole transformation.
	On the other hand we have seen  that t^ u(t)
		∈ L^∞(0,∞; L^p()) for all p∈[1,∞]. Therefore, the problem of the asymptotic behavior of solutions arises or, more precisely, the behavior of t^ u(t) in
		L^p()  when  t→∞.
		
		The following theorem shows that the asymptotic behavior is given by the self-similar solutions constructed in the previous section.
		
		
	Let u_0∈ L^1() with ∫_-∞^∞ u_0(x) dx=M. Let
			u=u(x,t) be the solution to (<ref>) and u_M=u_M(x,t) the self-similar solution of mass M obtained in (<ref>).
Then
			
    t^u(t)-u_M(t)_p→ 0,     when   t→∞

			for all p∈[1,∞].
		
		
		
			As we saw in (<ref>), u=w_x/w where w is the solution to the heat equation (<ref>).
			
	On the other hand u_M is of the form
			u_M=w_M,x/w_M with 
			
    w_M(x,t)=g_M(x/√(t))=(e^M-1)∫_-∞^x/√(t) h(s) ds+1,

			h being the gaussian defined in (<ref>).
			
			Thanks to(<ref>) we know that
			
    e^-|L|≤ w(x,t)≤ e^|L|,    ∀ (x,t)∈×(0,∞)

			and on the other hand, from the expression of w_M we can deduce
			
    e^-M^-≤ w_M(x,t) ≤ e^M^+

			with M^+=max(M,0) and M^- = -min(M,0).
			
			Combining the forms of u and u_x with (<ref>)–(<ref>) we see that (<ref>) is equivalent to
			
    t^w_x(t) w_M(t)-w(t) w_M,x (t)_p→ 0     when   t→∞

			which is the same as:
			
    t^(w_x(t)-w_M,x (t)) w_M(t)-(w(t) -
    					w_M(t)) w_M,x (t)_p→ 0

			when  t→∞.
			Thanks to (<ref>) we know that w_M ∈
			L^∞(×(0,∞)). On the other hand
			
    w_M,x=t^-1/2 (e^M-1) h(x√(t))

			satisfies t^ w_M,x (t) ∈ L^∞(0,∞; L^p()).
			
			It is therefore enough to prove that
			
    t^w_x(t)-w_M,x(t)_p    →0
    w(t)-w_M(t)_∞   →  0.

			
			We see that w_x is solution to the linear heat equation with
			mass e^M-1 as, for  (<ref>), we have
			
    ∫_-∞^∞ w_x(x,t) dx=lim_x→+∞ w(x,t) -
    			lim_x→-∞ w(x,t)=e^M-1.

			
			On the other hand, as w_M,x is the fundamental solution of the heat equation with mass e^M-1, (<ref>) is a consequence of Theorem 1.1.
			
			On the other hand,
			
    w(x,t)-w_M(x,t)=∫_-∞^x (w_x(s,t) - w_M,x
    			(s,t)) ds

			and
			
    lim_x→-∞ w(x,t)=lim_x→-∞ w_M(x,t)=1.

			Thus  (<ref>) is a consequence of (<ref>) for p=1.
		
		
		
			The equation
			
    u_t-u_xx = a(u^2)_x    in ×(0,∞)

			with a 1 can easily be reduced to the case a=1.
			
			Indeed, if u is solution to (<ref>) with initial data u_0, v=au is solution to			
    v_t-v_xx = (v^2)_x    in ×(0,∞)

			con initial data au_0.
			
			This allows us to calculate the self-similar solutions to (<ref>). Let
			f_M,a be the self-similar profile of the equation (<ref>) with mass M. 
		It holds f_M,a=(1/a) f_aM where f_aM is the self-similar profile of mass aM of the equation (<ref>).
			
			By virtue of (<ref>) we have 
			
    f_M,a(x)=(e^aM-1) h(x)/ a[(e^aM-1)
    					∫_-∞^x h(s) ds+1].

		
		
		

CHAPTER: THE HEAT EQUATION IN SIMILARITY VARIABLES

		
		

§ MOTIVATION: THE SELF-SIMILAR VARIABLES

		
		As we saw in the introduction, the convection-diffusion equation
		
    u_t-Δ u=a·∇(|u|^q-1u)    in ^N×(0,∞).

		can only admit self-similar solutions with profile in L^1(^N) si
		q=1+1/N. In this case we obtain the equation
		
    u_t-Δ u=a·∇(|u|^1/Nu)    in ^N×(0,∞)

	and the self-similar solutions are of the form
		
    u(x,t)= t^-N/2f(x/√(t))

		with a profile f that satisfies the elliptic equation
		
    -Δ f-x·∇ f/2-N/2 f=a·∇
    			(|f|^1/Nf)    in ^N.


In the study of self-similar solutions and asymptotic behavior, it is convenient to introduce new space-time variables so that the self-similar profiles become stationary solutions of the new evolution equation, and so that the self-similar asymptotic behavior of the general solution is equivalent to the convergence of the new trajectories to the self-similar profiles.

This point of view has been adopted by various authors in the study of different problems. Let us cite, among others, the works of M.   Escobedo and O.   Kavian <cit.> on the asymptotic behavior of parabolic equations with terms of absorption and the work of O.   Kavian <cit.> on the heat equation with a reaction term.
		
		In the case of the equation (<ref>) that concerns us, the similarity variables are
		
    y=x/√(t+1)  	s=log(t+1).

		If u is solution to (<ref>) with initial data u(0)=u_0, then
		
    v(y,s)=e^Ns/2u(e^s/2y,e^s-1)

		is solution to
		
    v_s-Δ v-y·∇ v/2-N/2=a·∇(|v|^1/N v)    en ^N×(0,∞)
    
    			 v(y,0)=u_0(y).

		In (<ref>), ∇ y
		Δ respectively represent the gradient and the Laplacian in the new spatial variable y.
		
		Reciprocally, if v=v(y,s) is solution to (<ref>), then
		
    u(x,t)=(t+1)^-N/2v(x/√(t+1),log(t+1))

		es solution to (<ref>) y
		
    u(x,t)-t^-N/2f(x√(t))_p→ 0     when  
    		t→∞

		if and only if
		
    v(s)→ f    in  L^p(^N)     when   s→∞.

		Thus, the existence of self-similar solutions for (<ref>)
		and the asymptotic self-similar behavior of its solutions transform into, respectively, the existence of stationary solutions for (<ref>) and the convergence of the latter's trajectories to equilibrium points.
		
		The equation (<ref>) is of parabolic type and the elliptic operator involved is
		
    Lv:=-Δ-y·∇ v/2.

		We observe that
		
    Lv=-1/K(y)÷(K(y)∇ v(y))

		where
		
    K(y)=exp(|y|^24).

		We see therefore that L is symmetric with respect to the scalar product
		
    (v,w)_K=∫ v(y)w(y)K(y) dy

		es decir, it holds
		
    (Lv,w)_K=(v,Lw)_K.

		therefore it is natural to introduce the weighted Sobolev spaces
		
    L^2(K)   ={ v∈ L^2(^N):∫|v(y)|^2K(y) dy<∞ }, 
    
    			H^m(K)   ={ v∈ H^m(^N):D^α v∈ L^2(K),∀α∈𝐍^N:|α|≤ m },     m∈

		endowed with the norms
		
    v^2_m,K   =∑_|α|≤ mD^αv^2_K,      m=1,2…
    v^2_K   =∫ |v(y)|^2 K(y) dy.

		These are Hilbert spaces.
		
		
		
		

§ ANALYSIS IN WEIGHTED SOBOLEV SPACES

		
		In this part we prove the fundamental properties of the operator L in the Sobolev spaces H^m(K). These results were shown by M. Escobedo and O. Kavian <cit.> and O. Kavian <cit.>.
		
		
			
				
  (i)
				It holds
				
    ∫|v(y)|^2|y|^2K(y) dy≤ 16∫ |∇ v(y)|^2 K(y) dy, 
    				    ∀ v∈ H^1(K).

				
				
  (ii)
				The inclusion H^1(K)⊂ L^2(K) is compact.
				
				
  (iii)
				L:H^1(K)→(H^1(K))^*
				is an isomorphism.
				
				
  (iv)
				L^-1:L^2(K)→ L^2(K) is compact.
				
				
  (v) The eigenvalues of L are positive real numbers.
				
    λ_k=N+k-1/2,     k=1,2,…

				
				The corresponding eigenspaces are				
    N(L-λ_kI)={ D^βφ_1:|β|=k-1 }

				with φ_1(y)=K^-1(y)=exp(-|y|^2/4).
				
				
  (vi)
				For all ϵ>0 and q≥1 there exists R>0 and a constant
				C=C(R,q,ϵ)>0 such that
				
    v^2_K≤ϵv^2_1,K+Cv^2_L^q(B_R),
    				    ∀ v∈ H^1(K)∩ L^q_ loc(^N).

			
		
		
		
			
				
  (i)
				Given v∈ H^1(K) set
				w=K^1/2v. It holds ∇ w-y/4
					w=K^1/2∇ v. As K^1/2∇ v∈(L^2(^N))^N We have ∇ w- y/4 w∈(L^2(^N))^N. 
				Besides,
				
    ∫|∇ w-y/4 w|^2   =∫(|∇ w|^2+
    					|y|^2/16|w|^2-w y·∇ w/2)
       =∫(|∇ w|^2+|y|^2/16|w|^2+N/4
    					|w|^2).

				In particular ∫|y|^2|w|^2=∫|y|^2|v|^2K, and therefore:
				
    ∫ |y|^2|v(y)|^2K(y) dy ≤ 16∫ |∇ v(y)|^2K(y) dy.

				
				
  (ii) Let v_n∈ H^1(K) be such that v_n→ v weakly in H^1(K)
				and we see that v_n-v_K→ 0. By virtue of (<ref>) it holds
				
    v_n-v^2_K   ≤exp(R^24)v_n-v^2_L^2(B_R)
    					+1/R^2∫|y|^2|v(y)|^2 K(y)
       ≤exp(R^24)v_n-v^2_L^2(B_R)
    					+16/R^2v_n-v^2_1,K.

				
				Given ϵ>0, since {v_n-v} is bounded in H^1(K), for R=R_0
				sufficiently large  it holds
				
    16/R^2_0v_n-v^2_1,K≤ϵ/2,
    				    ∀ n≥ 1.

				On the other hand, since the inclusion H^1(B_R_0)⊂ L^2(B_R_0)
				is compact, it holds
				
    v_n-v_L^2(B_R_0)≤ϵ/2,  for all  n≥
    				n_0

				if n_0 is sufficiently large.
			
				Combining (<ref>) with R=R_0, (<ref>) and (<ref>) we deduce  that v_n-v_K→ 0
				 when  n→∞.
				
				
  (iii)
				From the compactness of the injection H^1(K)⊂ L^2(K) we deduce  that 
				
    λ_1=min_{  v∈ H^1(K) 
     v0 }{∫ |∇ v|^2K/∫
    					|v|^2K}>0

				and therefore ∇ v_K and v_1,K are equivalent norms in H^1(K).
				
				As the bilinear form
				
    a(v,w)=(Lv,w)_K= (v,Lw)_K=∫∇ v∇ wK

				is coercive in H^1(K), by the Lax-Milgram Theorem we deduce that
				L:H^1(K)→(H^1(K))^* is an isomorphism.
				
				
  (iv)
				We firstly note that
				
    D(L)={ v∈ L^2(K): Lv∈ L^2(K) }=H^2(K).

				As
				
    (Lv,v)_K=∫|∇ v|^2K

				we deduce that D(L)⊂ H^1(K).
				
			Let v∈ D(L) be such that Lv=f∈ L^2(K) and define w=K^1/2v. As
				v∈ H^1(K) it holds w∈ H^1(^N) and also:
				
    -Δ w+(N/4+|y|^2/16)w=K^1/2
    				f∈ L^2(^N).

				On the other hand
				
    ∫|-Δ w+(N/4 +
    						|y|^2/16) w|^2
       =∫(|Δ w|^2 + N^2/16
    						w^2+|y|^4w^2/256+N|y|^2/32 w^2+N/2 |∇ w|^2
    						+|y|^2/8|∇ w|^2-N/8|w|^2)

				from where we deduce  that Δ w∈ L^2(^N) and therefore
				
    w∈ H^2 (^N).

				Besides, |y|^2 w, |y||∇ w|∈ L^2(^N), which implies
				
    |y|^2v∈ L^2(K),     |y||∇ v|∈ L^2(K).

				
				Combining (<ref>)–(<ref>) it can be shown that ∂_ij^2v∈
				L^2(K),∀ i,j ∈{1,…,N} which ensures that v∈
				H^2(K). It can easily be shown that  L is selfadjoint. In the part (iii)
				we saw that L is positive.
				
				On the other hand, L^-1:(H^1(K))^*→ H^1(K) is continuous and
				since the injections (H^1(K))^* ⊂ L^2(K) and
				H^1(K)⊂ L^2(K) are compact, we deduce that L^-1:L^2(K)→
				L^2(K) is compact.
				
				
  (v) As seen in the previous parts, L has a non-decreasing sequence of positive eigenvalues{λ_k}_k≥ 1 that goes to infinity  when  k→∞.
				
				Let λ>0 and v∈ L^2(K) be such that
				
    Lv=λ v.

				As |y|^m v∈ L^2(^N) for all m∈, we see that v̂ – the Fourier transform of v – satisfies v̂∈ H^m(^N) for all
				m∈ and therefore v̂∈ C^∞(^N).
				
	Applying the Fourier transform in the identity (<ref>) and applying the formula (y∇ v)(z) = -v̂(z) 
					+1/2 z·∇v̂(z) we obtain
				
    |z|^2 v̂(z)+N/2v̂(z) +1/2 z·∇v̂(z)
    				=λv̂(z)

				and therefore, w=e^|z|^2v̂ satisfies
				
    z·∇ w(z)=(2λ-N) w(z).

				It is the Euler's equation that is satisfied by the homogeneous functions of degree 2λ-N. As w∈ C^∞(^N) necessarily
				2λ-N∈. So we obtain the eigenvalues
				
    λ_k=N+k-1/2,     k=1,2,3,…

				
				The functions w_k solution to (<ref>) for λ=λ_k are 
				
    w_k(z)=P_k-1(z)

				where P_k-1 is a homogeneous polynomial of degree k-1 arbitrary,
				i.e., P_k-1(z)=z^β=z^β_1⋯ z_N^β_N for some
				β=(β_1,…,β_N)∈^N such that
				|β|=β_1+…+β_N=k-1. therefore
				
    v̂_k(z)=e^-|z|^2P_k-1(z)

				and applying the inverse Fourier transform we obtain
				
    v_k(y)=P_k-1(D)[e^-|y|^2/4]=D^β[e^-|y|^2/4]

				with |β|=k-1.
				
				
  (vi)
				Thanks to(<ref>) we have
				
    ∫|v(y)|^2K(y)    ≤exp(R^24) ∫_B_R |v(y)|^2
    					+1/R^2∫_B_R^c |v(y)|^2 K(y)
       ≤exp(R^24) ∫_B_R |v(y)|^2
    					+16/R^2∫|∇ v(y)|^2 K(y).

				
				Le tR>0 be such that 16/R^2=ϵ, So,
				
    v_K^2≤ϵ∇ v^2_K + e^R^2/4v_L^2(B_R)^2

				and as
				
    v_L^2(B_R)^2 ≤ C(R,q)v_L^q(B_R)^2

				for all q≥2, we deduce  (<ref>)  when q≥ 2.
				
				if 1≤ q<2, simply note that
				
    v_L^2(B_R)≤ Cv_L^q(B_R)^αv_H^1(B_R)^1-α

				for some 0<α<1. Therefore,
				
    v_L^2(B_R)^2≤ϵv_H^1(K)^2 + C(ϵ)
    					v_L^q(B_R)^2.

				We deduce (<ref>) for all 1≤ q<2.
			
		
		
		

§ THE HEAT EQUATION IN SIMILARITY VARIABLES

		
		The linear heat equation
		
    u_t-Δ u= 0    en ^N×(0,∞)
    
    			u(x,0)= u_0(x)   en ^N

		with the change of variables (<ref>) transformes into
		
    v_s-Δ v-y·∇ v/2-N/2v=0     en
    				^N×(0,∞)
    
    			v(y,0)=u_0(y)

		The parabolic operator of this system		
    ∂/∂ s+L-N/2 I

		coincides with the linear part of the nonlinear equation (<ref>).
		
		It is thence interesting to study the asymptotic behavior of the solutions to (<ref>).
		
		Let {φ_l}_l≥ 1 be an orthonormal basis of L^2(K) consisting of eigenfunctions of L. Every eigenfunction φ_l
		corresponds to the eigenvalue μ_l of the operator L, such that
		{μ_l}_l≥ 1 is a non-decreasing sequence of eigenvalues in which each eigenvalue N+k-1/2 appears as many times as its multiplicity indicates. In particular
		
    μ_1=N/2,    μ_2=N+1/2

		and the first eigenfunction φ_1 is
		
    φ_1(y)=(4π)^-N/4exp(-|y|^2/4).

		Let u_0∈ L^2(K). Let {α_l}_l≥ 1 be its Fourier coefficients	
		
    α_l=(u_0,φ_l)_K=∫ u_0(y)φ_l(y)K(y) dy

		so that 
		
    u_0=∑_l≥1α_l φ_l.

		
		We observe that
		
    α_1=(u_0,φ_1)_K=(4π)^-N/4∫ u_0(y) dy.

		
		The solution v=v(y,s) to (<ref>) is given by the formula
		
    v(y,s)   =∑_l≥1 e^-(μ_l-N/2)sα_lφ_l(y)
    			=α_1φ_1(y) +ṽ(y,s) 
     with 
    ṽ(y,s)   =∑_l≥2 e^-(μ_l-N/2)sα_lφ_l(y)

		Taking into account that if ψ=∑_l≥1β_lφ_l,
		
    ψ_m,K^2=∑_l≥1(μ_l)^m β_l^2

		from (<ref>) we deduce that
		
    v∈ C([0,∞); L^2(K))∩ C^∞((0,∞);
    		H^m(K))

		for all m∈. (This is the usual regularizing effect in semigroups generated by selfadjoint operators.)
		
		In the decomposition  (<ref>) also observe that the projection of the solution v(s) on the first eigenspace E_1={φ} does not depend on s, i.e.,
		
    ∫ v(y,s) dy=∫ u_0(y) dy,    ∀ s≥ 0.

		Hence, the mass of the solution is conserved.
		
		On the other hand
		
    μ_l-N/2≥1/2,     ∀ l≥ 2

		and therefore for the component ṽ(y,s) we obtain an exponential decay:
		
    ṽ(s)_K≤ e^-s/2ũ_0_K,    ∀ s≥ 0

	 ũ_0 being the projection of u_0 on E_1^⊥:
		
    ũ_0(y)=∑_l≥2α_lφ_l(y).

		
		Therefore,
		
    v(s)-α_1φ_1_K ≤ e^-s/2ũ_0_K,     ∀ s≥1

		for all m∈.
		
		It is also easily checked that
		
    v(s)-α_1 φ_1_m, K≤ C_m e^-s/2ũ_0_K,      s ≥ 1

		for all m ∈.
		
		The operator L-N/2I generates an analytic semigroup of contractions in L^2(K) that we henceforth denote by {S(s)}_s≥0
		which is such that
		
    S(s) u_0=v(s),    ∀ u_0∈ L^2(K),∀ s>0

		 v=v(s) being the solution to (<ref>).
		
		The following result then holds.
		
		
			
				
  (i)
				The linear parabolic equation (<ref>) admits a unique solution
				v(s)=S(s)u_0∈ C([0,∞); L^2(K)) for every
				initial data u_0∈ L^2(K). This solution satisfies
				
    v∈ C^∞((0,∞);H^m(K)),    ∀ m∈.

				
				
  (ii)
				The projection of the solution v on the first eigenspace E_1 remains invariant				
    (v(s),φ_1)_K=(u_0,φ_1)_K=α_1,    ∀
    					s≥ 0.

				
				
  (iii)
				It holds
				
    v(s)_K≤ Ce^-s/2u_0_K,    ∀ s≥0,∀ u_0∈
    				E_1^⊥

				and for all m∈ there exists C_m>0 independent of initial data such that 
				
    v(s)_m,K≤ C_m e^-s/2u_0_K,    ∀ s≥1,∀
    					u_0∈ E_1^⊥.

				
				
  (iv)
				On the other hand
				
    (L-N/2) v(s)_K ≤1/√(2)su_0_K,    ∀ s>0, ∀ u_0∈ E_1^⊥

				y
				
    ∇ v(s)_K^2-N/2v(s)_K^2 ≤1/2su_0_K^2,    ∀ s>0,
    					∀ u_0∈ E_1^⊥

			
		
		
		
			The propositions (i)–(iii) have previosly been proven. Since
			L-N/2I is a maximal monotone and selfadjoint operator on 
			L^2(K) with domain H^2(K), (<ref>) and (<ref>) are a consequence of the regularizing effect of the equation (cf. T. Cazenave and A. Haraux <cit.>, Th.
			3.2.1.)
		
		
		As H^m(K)⊂ W^m,p(^N) with continuous injection for all
		p∈[1,2], from (<ref>) we deduce  that
		
    v(s)-α_1φ_1_W^m,p(^N)≤ C_m e^-s/2u_0_K,
    		    ∀ s≥1.

		
		On the other hand, for all m∈ and p∈[2,∞] there exists m'∈
		such that H^m'(^N)⊂ W^m,p(^N) with continuous injection and thus, we deduce that (<ref>) es cierto for all m∈ y
		p∈[1,∞].
		
		In terms of the solution u=u(x,t) of the heat equation (<ref>), the estimate (<ref>) implies 
    D^α u(x,t)-MD^α G(x,t)_p ≤ C_p,m
    		t^--(|α|+1)/2,    ∀ t≥1

		for all α∈ and p∈[1,∞], M  being the mass of the
		solution: M=∫ u_0(x) dx. In particular we obtain
		
    u(t)-MG(t)_p ≤ C_p t^--(|α|+1)/2,
    		    ∀ t≥1

		for all p∈[1,∞].
		
		We obtain therefore a result analogous to the proof of Lemma 1.2 of Chapter 1, but this time only for initial data u_0∈ L^2(K). Nevertheless, since L^2(K) is dense in L^2(^N), the density argument
		used in the proof of Theorem 1.1 of Chapter I now allows us to conclude that
		
    t^u(t)-MG(t)_p→0     when    t→∞

		for all p∈[1,∞] and u_0∈ L^1(^N) such that ∫
			u_0(x) dx=M.
		
		Thus, by means of the self-similar variables, we obtain optimal results about the asymptotic behavior of the solutions to the heat equation.
		
		
		

CHAPTER: THE CAUCHY PROBLEM

		
		

§ THE CAUCHY PROBLEM IN L^1(^N)∩ L^∞(^N)

		
		In this part, we study the equation
		
    u_t-Δ u=a·∇(F(u))     in ^N×(0,∞),
    
    			u(0)=u_0

	with initial data u_0∈ L^1(^N)∩ L^∞(^N).
		
		We have the following results of existencia, uniqueness and regularity of
		solutions.
		
			Suppose that F∈ C^1() and F(0)=0. Then, for all initial data u_0∈ L^1(^N)∩ L^∞ (^N) there exists a unique
			solution u∈ C([0,∞);L^1(^N))∩
			L^∞(^N×(0,∞ )). This solution
			satisfies u∈ C((0,∞);W^2,p(^N))∩
			C^1((0,∞ );L^p(^N)) for all p∈(1,∞).
			
			Besides, if u and v are solutions to (<ref>) corresponding to initial data u_0, v_0 ∈ L^1(^N)∩
			L^∞(^N), it holds
			
    u(t)-v(t)_1≤u_0-v_0_1,    ∀ t≥ 0.

			
			In other words, (<ref>) a contraction semigroup in L^1(^N).
		
		
		
			
			
				
  (1) Existence.
				
				We consider the integral equation associated to (<ref>):
				
				
    u(t)   =G(t)*u_0+∫^t_0 G(t-s)*a·∇(F(u(s))) ds 
       =G(t)*u_0+∫^t_0 a·∇ G(t-s)*F(u(s)) ds

				with G being the heat kernel
				
    G(x,t)=(4π t)^-N/2exp(-|x|^2/4t)

			and we henceforth denote by the símbolo * the convolution in the spatial variables.
				
				The solutions to (<ref>) are weak solutions of (<ref>). We seek  the  local solution to (<ref>) (for 0≤ t≤ T, with T>0 sufficiently small) as a fixed point of the operator 
    [ϕ(u)](t)=G(t)*u_0
    				+∫^t_0a·∇ G(t-s)*F(u(s)) ds

				in the Banach space
				
    X_T=C([0,T];L^1(^N))∩
    				L^∞(^N×(0,T))

				with T>0 sufficiently small that we will choose later. We will apply the Banach fixed point theorem. It will be enough to prove that ϕ is a strict contraction in B_R, the ball of radius R in X_T:
				
    B_R={ u∈ X_T:sup_t∈
    					[0,T]{u(t)_1+u(t)_∞}<R }

				with a radius R that we will choose later.				
				Let M=max{u_0_1,u_0_∞}. Let
				r=1,∞ and let us estimate the L^∞(0,T;L^r(^N)) norm of ϕ(u) for u∈ X_T.
				Thanks to the Young inequality it holds
				
    [ϕ(u)](t)_r≤G(t)_1 u_0_r+∫^t_0 a·∇
    				G(t-s)_1 F(u(s))_r ds.

				
				By means of an explicit computation it can be seen that				
    G(t)_1   = 1,∀ t>0
    ∇ G(t)_1   ≤ Ct^-1/2,∀ t>0.

				therefore
				
    [ϕ(u)](t)_r≤ M+C|a|∫^T_0(t-s)^-1/2F(u(s))_r ds.

				On the other hand
				
    F(u(t))_r≤ m(R)u(t)_r,    ∀ t∈ [0,T]

				con
				
    m(R)=max_z∈ [0,R]|F(z)/z|

				y
				
    u(t)_r<R,    ∀ t∈ [0,T]

				for r=1,∞, for all u∈ B_R.
				
				Thus
				
    [ϕ(u)](t)_r   ≤ M+C|a|m(R)R∫^t_0(t-s)^-1/2 ds
       =M+2C|a|m(R)Rt^1/2
       ≤ M+2C|a|m(R)RT^1/2,    ∀ t∈ [0,T]

				for r=1,∞.
				
				If R, T>0 are such that				
    M+2C|a|m(R)RT^1/2<R/2

				we will have guaranteed that ϕ(B_R)⊂ B_R.
				
				For (<ref>) to be fulfilled, we choose R such that
				
    R≥ 4M

				and then T>0 sufficiently small as to have
				
    2C|a|m(R)RT^1/2<R/4.

				We now estimate ϕ(u)-ϕ(v) in L^∞(0,T;L^r(^N))
				for all u,v∈ B_R. It holds
				
    [ϕ(u)](t)-[ϕ(v)](t)_r    ≤∫^t_0 a·∇
    					G(t-s)_1 F(u(s))-F(v(s))_r ds
       ≤ C|a|∫^t_0(t-s)^-1/2F(u(s))-F(v(s))_r ds

				
				Set
				
    L(R)= max_z∈ [0,R] |F'(z)|.

				
				Then
				
    [ϕ(u)](t)-[ϕ(v)](t)_r   ≤ C|a|L(R)∫^t_0(t-s)^-1/2u(s)-v(s)_r ds
       ≤ 2C|a|L(R)T^1/2u-v_L^∞(0,T;L^r(^N)),
    ∀ t∈ [0,T],∀ u,v∈ B_R.

				
				To guarantee that
				ϕ is a strict contraction in B_R it will be enough to choose
				T sufficiently small as to, in addition to (<ref>), 
				
    2C|a|L(R)T^1/2<1

				holds as well.				
				In this way we deduce that ϕ admits a unique fixed point in B_R,
				which ensures the existence of a local solution of the integral equation (<ref>) for t∈[0,T]. This solution satisfies (<ref>) for t∈
				[0,T].
				
				By means of classical extension methods (see for instance
				<cit.>), we deduce  the existence of a maximal solution
				u∈ C([0,T_m);L^1(^N))
				∩ L^∞(^N×(0,T_m))
				where T_m>0 is the maximal time of existence that satisfies the
				following alternative:
				
    T_m=∞

				or
				
    Si T_m<∞,  lim sup_t→ T^-_m{u(t)_1+u(t)_∞}=∞.

				In order to ensure that the solution is global, that is to say that
				u∈ C([0,∞); L^1(^N))∩
				L^∞(^N×(0,∞)), it is enough to prove a priori estimate for the solution, meaning
				
    sup_t∈[0,T]{u(t)_1+u(t)_∞}<∞.

				
				As mentioned in what precedes, the solution to (<ref>) is a solution of
				the differential equation  (<ref>) in ^N×[0,T_m).
				Classical results on the
				regularity for the linear heat equation and a	boot-strap argument allows for showing that the constructed solution satisfies the following regularity:
				
    u∈ C((0,T_m);W^2,p(^N))∩
    					C^1((0,T_m);L^p(^N)).

				
				Multiplying formally the equation (<ref>) by
				
    (u) = 
    				1   u>0  
      0   u=0  
      -1   u<0

				and integrating in ^N we obtain
				
    d/dt∫ |u(x,t)| dx
    						-∫Δ u(x,t)(u(x,t)) dx
    
    						=∫ a·∇(F(u(x,t))(u(x,t)) dx
    
    						=∫ a·∇(H(u(x,t))) dx

				with
				
    H(z)=∫^z_0 F'(s)(s) ds.

				Integrating by parts we deduce  that
				
    ∫ a·∇(H(u)) dx=0

				and the Kato inequality ensures that
				
    -∫Δ u(u) dx≥ 0

				so
				
    d/dt∫ |u(x,t)| dx≤ 0

				and therefore
				
    u(t)_1≤u_0_1,    ∀ t∈ [0,T_m).

				To justify the formal calculation that we have just carried out, we define Lipschitz regularization of (u):
				
    θ_ϵ(u)=
    					u/ϵ   |u|<ϵ
    (u)   ϵ≤|u|≤1/ϵ
    
    					u-(1/ϵ-1)   |u|≥1/ϵ.

				As u∈ L^1(^N)∩ L^∞(^N) it can be seen that
				θ_ϵ(u) ∈ L^1(^N)∩ L^∞ (^N) for all
				ϵ >0, and 
				
    θ_ϵ(u)→(u)    a.e.    x∈^N, ∀
    				t∈ [0,T_m).

				
				Multiplying the equation (<ref>) by θ_ϵ(u) and integrating
				in ^N we now obtain in a rigorous way that
				
    ∫ a·∇(F(u))θ_ϵ(u) dx   =0
    ∫Δ uθ_ϵ(u) dx   ≥0

				for all ϵ >0 and integrating in [0,t] with respect to time, we obtain
				
    ∫^t_0∫ u_t(x,s)θ_ϵ(u(x,s)) dx ds   =∫^t_0(d/dt∫γ
    					_ϵ(u(x,s)) dx) ds
       =∫γ_ϵ(u(x,t)) dx
    					-∫γ_ϵ(u_0(x)) dx ≤ 0

				where
				
    γ_ϵ(z)=∫^z_0θ_ϵ(s) ds.

				
				Thus
				
    ∫γ_ϵ(u(x,t)) dx≤∫γ_ϵ(u_0(x)) dx.

				Passing to the limit in this inequality  when  ϵ→ 0, by the Lebesgue dominated convergence Theorem we obtain (<ref>).
				
	To estimate the L^∞ norm of the solution we define 
				μ=u_0_∞. We multiply the equation (<ref>) by
				[(u-μ)^+] where
				
    z^+=
    					
    					z   z≥0  
      0   z≤0

				Integrating in ^N and by arguments analogous to those used in what precedes, we deduce that 
				
    d/dt∫(u(x,t)-μ)^+ dx≤ 0.

				As (u_0-μ)^+=0 we obtain that
				
    u(x,t)≤μ, p.c.t. x∈^N, ∀ t∈ [0,T_m).

				
				Multiplying the equation by [(u+μ)^-] with z^-=-(-z)^+,
				we obtain
				
    u(x,t)≥ -μ , p.c.t. x∈^N, ∀ t∈ [0,T_m).

				we obtain
				
    u(t)_∞≤μ,    ∀ t∈ [0,T_m).

			From (<ref>) and (<ref>) we deduce  that T_m=∞, which concludes the proof of the existence of a global solution. From (<ref>) we obtain the regularity of the solution stated in the Theorem.
				
				
  (2) Uniqueness.
				
				The uniqueness is an immediate consequence of the following L^1 contraction property enjoyed by the semigroup generated by the system (<ref>):
				
				
    u(t)-v(t)_1 ≤u_0-v_0_1,    ∀ t>0

				for all solutions to (<ref>) with initial data u_0,v_0∈
				L^1(^N)∩ L^∞ (^N).
				
				To prove this property, simply multiply by (u-v) the equation satisfied by u-v and integrate in ^N. we obtain
				
    d/dt(u-v)(t)_1≤ 0.

				Again, to justify this calculation we multiply the equation by
a regularization θ_ϵ(u-v) and we pass to the limit as
				ϵ→0. 
			
		
		
		

§ DECAY IN L^P(^N)

		
		In this part we will obtain L^p estimates for the solutions to (<ref>). We will show decay results s analogous to those obtained in Chapter I for the linear heat equation.		
		The following result holds.
		
		
			For all p∈ [1,∞] there exists a positive constant C=C(p,N)>0 such that
			
    u(t)_p≤ Cu_0_1 t^-,    ∀ t>0

			for all solution to (<ref>) with initial data u_0∈ L^1(^N)∩
			L^∞ (^N).
		
		
		
			
				
  (1)
				The estimate (<ref>) analogous to the one we obtain for solutions to the linear heat equation. This estimate implies that
				t^u(t)∈ L^∞(0,∞ ;L^p(^N)).Therefore, its behavior in L^p(^N) should be considered when t→+∞.
				
				
  (2)
				The estimates will be demonstrated by means of the multiplier method.
				We multiply the equation by |u|^r-1u for any r>1.
				Integrating on ^N, we see that the convective term vanishes. That is why we obtain the same estimates that for the heat equation linear. An iterative argument allows obtaining the results for
				p=∞. In this case (<ref>) means
				
    u(t)_∞≤ Cu_0_1t^-N/2,    ∀ t>0.

				
				
  (3)
	These estimates are very useful when extending the results of existence and uniqueness of solutions to (<ref>) to initial data u_0∈ L^1(^N).
			
		 
		
		
			The case p=1 is an immediate consequence of (<ref>), which provides the estimate
			
    u(t)_1≤u_0_1,    ∀ t≥ 0.

			
			In the case p∈ (1,∞), we multiply the equation (<ref>) by
			|u|^p-2u and integrate in ^N. We obtain
			
    1/pd/dt∫ |u(x,t)|^p dx+(p-1)∫|∇
    					u(x,t)|^2|u(x,t)|^p-2 dx
    
    					=∫ a·∇(F(u(x,t)))
    					|u(x,t)|^p-2u(x,t) dx
    
    					=∫ a·∇(H_p(u(x,t))) dx

			with
			
    H_p(z)=∫^z_0 F'(s)|s|^p-2s ds

			By the arguments of the proof of Theorem 4.1 we obtain
			
    ∫ a·∇(H_p(u(x,t))) dx=0

			and therefore
			
    1/pd/dt∫ |u(x,t)|^p dx+ 4(p-1)/p^2∫|∇ (|u(x,t)|^p/2)|^2 dx=0

			We need the following interpolation lemma.
			
			
				For all p∈(1,∞) there exists a constant
				C=C(p,N)>0 such that
				
    v_p^(N(p-1)+2)p/N(p-1)≤
    					Cv^2p/N(p-1)_1∇ (|v|^p/2)^2_2

				For all
				v∈ W^2,p(^N)∩ L^1(^N).
			
			
			
				Note that since v∈ W^2,p(^N), entonces Δ v∈ L^p(^N).
				therefore the integral
				
    ∫Δ v|v|^p-2v dx

				is well defined. By Green's formula it is observed that it coincides with				∫ |∇ (|v|^p/2)|^2 dx, from where we deduce  that ∇
				(|v|^p/2)∈ L^2(^N). All the terms in (<ref>) are thence well defined.
			 
			
			
				We will just describe the main steps of the proof. For more details, see <cit.>.				
				Let us consider solely the case N≥ 3. By the Sobolev inequality
				(cf. H. Brezis <cit.>, Th. IX.9) we know that there exists a constant
				C_N>0 (which depends only on the dimension N) such that
				
    v_2N/(N-2)≤ C_N∇ v_2,∀ v∈ H^1(^N).

				therefore
				
    v^p/2_Np/(N-2)=|v|^p/2_2N/(N-2)≤
    				C_N∇ (|v|^p/2)_2.

				So, to obtain (<ref>), it will suffice to prove
				
    v^(N(p-1+2)p/N(p-1)_p≤
    				Cv^2p/N(p-1)_1v^p_Np/(N-2)

				This is an immediate consequence of the classical interpolation inequalities in L^p (see, for instance, H. Brezis
				<cit.>, Ch. IV).
			 
			
			

  
*End of proof of Theorem 4.2:
			Combining (<ref>) and (<ref>) we obtain
			
    d/dt(u(t)^p_p)+4(p-1)/Cpu(t)_p^(N(p-1)+2)p/N(p-1)/u(t)^2p/N(p-1)_1≤ 0.

			Combining (<ref>) and (<ref>) we obtain
			
    d/dt(u(t)^p_p)+4(p-1)/Cpu_0^-2p/N(p-1)_1
    			u(t)^(N(p-1)+p/N(p-1)_p ≤ 0.

			
			It can easily be shown that the solutions φ≥0 of the differential inequality
			
    φ'(t)+α(φ(t))^β≤ 0,    ∀ t>0

			with α>0, β>1 satisfy
			
    φ(t)≤(α (β -1)t
    			+φ^1-β(0))^-1/β-1,    ∀ t>0

			and therefore, in particular,
			
    φ(t)≤(α (β-1)t)^1/β-1,
    				    ∀ t>0.

			
			From (<ref>)–(<ref>) we deduce 
			
    u(t)^p_p≤[8u_0^-2p/N(p-1)_1/NCpt]^-N/2 (p-1)

			and therefore
			
    u(t)_p≤u_0_1(8NCp)^-t^-.

			This concludes the proof of (<ref>) for p∈ [1,∞ ).
			
			Note that the constant
			
    C_p=(8NCp)^-

			from the estimate (<ref>) does not remain bounded when  p→∞ . It is for this reason that in the case p=∞ the estimate (<ref>) is not obtained by passing to the limit when  p→∞. It is necessary to obtain L^q-L^p estimates and use an iterative argument.
			
			The interpolation inequality (<ref>) for p=2 imples
			
    v^2(N+2)/N_2≤ Cv^4/N_1∇ v^2_2

			for all v∈ H^2(^N)∩ L^1(^N). By density, this inequality
			extends to all v∈ H^1(^N)∩ L^1(^N).
			
			Combining (<ref>) and (<ref>) with p=2q for q>1 we obtain
			
    d/dt(u(t)^2q_2q)+
    				2(2q-1)/Cqu(t)^2q(N+2)/N_2q/u(t)^4q/N_q≤ 0.

			On the other hand, multiplying the equation (<ref>) por
			|u|^q-2u we easily obtain that
			
    d/dt∫ |u(x,t)|^q dx≤ 0.

			and therefore
			
    u(t)_q≤u_0_q,    ∀ t>0

			for all q∈[1,∞].
			
			Combining (<ref>)–(<ref>) 	it follows that		
    d/dt(u(t)^2q_2q)+
    				2(2q-1)/Cqu_0^4q/N_qu(t)^2q(N+2)/N_2q≤
    				0.

			
		Applying estimate (18) which is satisfied by solutions to (17), it follows that			
    u(t)^2q_2q≤[4(2q-1)/NCqu_0^4q/N_q
    			t]^-N/2

			and thus
			
    u(t)_2q≤(NCq4(2q-1))^N/4qu_0_q t^-N/4q,    ∀ t>0.

			As q/(2q-1)≤ 1 we deduce  that
			
    u(t)_2q≤(NC4)^N/4qu_0_q t^-N/4q,    ∀ t>0.

			As the equation (<ref>) is invariant with respect to translations in  t, from (<ref>) we deduce that
			
    u(t+s)_2q≤(NC4)^N/4qu(t)_q s^-N/4q,    ∀ s,t>0.

			Given τ>0, choosing s=τ 2^-(n+1) and q=2^n we obtain
			
    u(t+τ 2^-(n+1))_2^n+1≤(NC4τ)^N2^-(n+2)
    			2^-N(n+1) 2^-(n+2)u(t)_2^n.

			
			By iterating we obtain
			
    u(τ(2^-1+… 
    			+2^-n))_2^n+1≤ C_nu_0_1

			with
			
    C_n=(NC4τ)^N ∑^n-1_j=0
    				2^-(j+2)(2^-N/2)^∑^n-1_j=0(j+1)2^-(j+1)

		and since 2^-1+…+2^-n≤ 1, por (<ref>) it holds
			
    u(τ)_2^n+1≤ C_nu_0_1.

			
			It can easily be shown that
			
    C_n→ 2^-N(NC4τ)^N/2  when  n→∞

			and therefore from (<ref>) we deduce 
			
    u(τ)_∞≤(NC4τ)^N/22^-Nu_0_1.

			This concludes the proof of (<ref>) for p∈[1,∞].
		
		
		
			Estimates of the type (<ref>) were proven by M. Schonbeck
			<cit.> for less general non-linearities F and more regular initial data using the Fourier transform. The estimate (<ref>) for
			p∈[1,∞] was shown in <cit.>. The proof of (<ref>) in the 
			case p=∞ is an adaptation of that done by L. Veron <cit.>
		for abstract semilinear parabolic equations.
		
		
		

§ THE CAUCHY PROBLEM IN L^1(^N)

		
		
			Let F∈ C^1() be such that F(0)=0. Then for all initial data u_0∈ L^1(^N) there exists a unique solution u∈
			C([0,∞); L^1(^N))∩ L^∞_ loc(0,∞
			;L^∞(^N)) to (<ref>)  such that
			
    u∈ C((0,∞ );W^2,p(^N))∩
    			C^1((0,∞);L^p(^N))

			for all p∈(1,∞).
			
			The solution satisfies (<ref>) for all p∈[1,∞].
			
			Moreover, if u and v are solutions to (<ref>) with
			initial data u_0, v_0 ∈ L^1 (^N) it holds
			
    u(t)-v(t)_1≤u_0-v_0_1,    ∀ t≥ 0.

		If u_0≥ v_0 it holds
			
    v(x,t)≤ u(x,t),     ∀ t>0, p.c.t. x∈^N.

		
		
		
			Let u_0∈ L^1(^N). We construct an approximation			{u_0,n}⊂ D(^N) such that
			
    u_0,n→ u_0  in  L^1(^N)  when  n→∞.

			By Theorem 4.1 we know that for every n∈ there exists a solution
			u_n∈ C([0,∞);L^1(^N))∩ L^∞(^N×(0,∞)) to (<ref>) with initial data u_0,n. By the  L^1(^N)  contraction
			property  (<ref>) it holds
			
    u_n(t)-u_k(t)_1≤u_0,n-u_0,k_1,     ∀ t>0,
    			∀ n,k∈.

			
			As {u_0,n} is a Cauchy sequence in L^1(^N), {u_n}
			is a Cauchy sequence in C([0,∞);L^1(^N)) and thus there exists u∈ C([0,∞ );L^1(^N)) such that
			
    u_n→ u  in  L^∞(0,∞;L^1(^N)),    when     n→∞.

			
			Obviously u(0)=u_0. We still need to prove that u satisfies the differential equation (<ref>). As u_n satisfies it, we proceed in passing to the limit as n→∞. The linear terms do not manifest a difficulty.
			Nevertheless (<ref>) is not sufficient for passing to the limit in the nonlinear term a·∇(F(u_n)) (except in the case when F is globally Lipschitz, in which, F(u_n
			(t))→ F(u(t)) in L^1(^N) and a·∇(F(u (t)))→ a·∇(F(u(t)) en
			D'(^N) for all t≥ 0). With the goal of solving this dififculty,  we observe that, thanks to  the estimate (<ref>), for all
			t>0 it holds
			
    {u_n(t)} is bounded in  L^∞ (^N).

			As F is locally Lipschitz, Combining (<ref>) and (<ref>) we deduce
			that
			
    u(t)∈ L^p(^N)      for all p∈[1,∞)
    
    			u_n(t)→ u(t)      en L^p(^N)  when  n→∞, ∀
    				p∈[1,∞)
    
    			F(u_n(t))→ F(u(t))      en L^p(^N)
    				 when  n→∞, ∀ p∈[1,∞)

			for all t>0.This allows us firstly to pass to the limit in (<ref>) and extend these estimates to the limit function u and secondly, pass to the limit in the equation (<ref>) to the deduce that that the limit u is solution (<ref>).
			
			Again, for classic regularity results for the linear heat equation and by means of  boot-strap bootstrap argument. We deduce that
			
    u∈ C((0,∞ );W^2,p(^N))∩ C^1((0,∞);
    			L^p(^N))

			for all p∈(1,∞).
			
			The uniqueness of the solution is once again an immediate consequence of the L^1 contraction property (<ref>) which can be shown as in Theorem 4.1.
			
			We finish the proof of Theorem 4.3 by using the basic comparison principle. Let u_0 and v_0∈ L^1(^N) be such that
			u_0(x)≥ v_0(x), a.e. x∈^N. Let u and v be the corresponding solutions to
			(<ref>).
			
			Multiplying by (u-v)^- the equation satisfied by u-v and
			integrating in ^N we obtain
			
    d/dt∫_^N(u(x,t)-v(x,t))^- dx≤ 0.

			As (u_0-v_0)^-=0, we deduce that (u-v)^-=0, which implies u≥
			v.
		 
		
		

§ GRADIENT ESTIMATES

		
		In this part, we obtain estimates for the gradient of the
		solutions to (<ref>) in the case where
		
    |F'(s)|≤ C|s|^1/N,    ∀ s∈.

		The following result holds. 
		
			Let F∈ C^1() be such that F(0)=0 and satisfies (<ref>). 
			So, for all u_0∈ L^1(^N) the solution u=u(x,t)
			to (<ref>) satisfies
			
    ∇ u(t)_p ≤ C_p t^--1/2,    ∀ t>0

			for all p∈[1,∞].
		
		
		
			Let us introduce the rescaled functions 
			
    u_λ(x,t)=λ^N u(λ x,λ^2 t).

			These functions satisfy
			
    u_λ,t-Δ u_λ   =λ^N+1 a·∇ 
    			F(λ^-N u_λ)
    
    			u_λ(0)=u_0,λ   =λ^N u_0(λ x)

			and therefore, for any τ>0 satisfy the integral equation
			
    u_λ(t+τ)= G(t)*u_λ (τ)+λ^N+1∫^t_0∫^t_0 G(t-s)*a·∇ F(λ^-N
    				u_λ(s+τ)) ds.

			Taking gradients we obtain
			
    ∇ u_λ(t+τ)=∇
    					G(t)*u_λ(τ)
    +λ∫^t_0 ∇ G(t-s)*F'(λ^-N
    					u_λ (s+τ))a·∇ u_λ(s+τ) ds.

			
			Since
			
    ∇ G(t)_p ≤ C_p t^--1/2,    ∀ t>0

			for all p∈[1,∞] and u_λ(t)_1≤u_0,λ_1=u_0_1 for all λ >0, t>0, taking
			 L^p norms in (<ref>) we obtain
			
    ∇ u_λ (t+τ)_p≤
    					C_pu_0_1 t^--1/2
    
    				+λ |a|∫^t_0 ∇ G(t-s)_1F'(λ
    					^-N u_λ (s+τ))_∞∇
    					u_λ(s+τ)_p ds.


			From (<ref>) we deduce that
			
    u_λ (t)_p ≤ C_pu_0_1 t^-,    ∀ t∈
    			(0,1)

			for all p∈[1,∞] with C_p>0 independent of  λ. Thus
			
    u_λ (s+τ)_∞≤ C_∞u_0_1τ^-N/2,    ∀ s≥ 0.

			
			Combining (<ref>), (<ref>), (<ref>) and (<ref>) we obtain
			
    ∇ u_λ (t+τ)_p≤
    					C_pu_0_1 t^--1/2
    
    				 +C|a|u_0^1/Nτ^-1/2∫^1_0 (t-s)^-1/2∇
    					u_λ (s+τ)_p ds.

			
			We need the following Gronwall Lemma.			
			
			Let T>0, A,B>0, α∈[0,1] and φ∈ C([0,T])
				a non-negative function such that
				
    φ (t)≤ At^-α+B ∫^t_0 (t-s)^-1/2φ
    					(s) ds.

				
				Then, there exists C>0 that depends only on T,α and B such that
				
    φ (t)≤ CA t^-α,     ∀ t∈(0,T].

			
			
			
				Multiplying in (<ref>) by t^α we obtain
				
    t^αφ (t)≤ A+Bt∫^t_0 (t-s)^-1/2s^-αψ
    					(s) ds

				with ψ(s)=sup_t∈[0,s]{t^αφ(t)}. Since
				ψ is non-decreasing it holds
				
    t^αφ (t) ≤ A+B^αψ (t)
    				∫^t_0(t-s)^-1/2s^-α ds=A+B C_α t^1/2ψ (t)

				con C_α=∫^1_0(1-s)^1/2s^-α ds. Taking supremums in
				this inequality we see that
				
    ψ(t)≤ A+B C_α t^1/2ψ (t).

				therefore, if t∈[0,t_0] with t_0=(2BC_α)^-2, it holds
				
    ψ (t)≤ 2A,     ∀ t∈[0,t_0].

				If T≤ t_0, (<ref>) is obtained from (<ref>). Si T>t_0, take δ≤
				t_0. Decomposing the integral in (<ref>) in the segments [0,δ],
				[δ,t-δ] and [t-δ,t] we obtain
				
    t^αφ (t)   ≤
    				A+BT^αδ^-α-1/2∫^t_0ψ(s) ds
         +Bt^αψ(t){∫^δ_0(t-s)^-1/2s^-α ds
    				+∫^t_t-δ(t-s)^-1/2s^-α ds}
       =A+BT^αδ^-α-1/2∫_0^tψ(s) ds
          +BT^α{(t_0-δ)^-1/2δ^1-α/(1-α ) +2(t_0-δ)^-αδ^1/2}ψ(t).

				
				Choosing δ>0 sufficiently small such that
				
    BT^α{(t_0-δ )^-1/2δ^1-α/1-α
    				+2(t_0-δ)^-αδ^1/2}≤1/2

				and taking supremums in this last inequality we obtain
				
    ψ (t)≤ 2(A+BT^αδ^-α-1/2∫^t_0ψ(s) ds).

				
				Applying the classic Gronwall Lemma and using (4.41) we obtain
				
    ψ(t)≤ 2A+4ABT^αδ^-α-1/2t_0exp(2BT^αδ^-α-1/2(t-t_0)),     t∈[t_0,T].

				This concludes the proof of (4.38).
			
			
			

  
*End of the proof of Theorem 4.4
			Applying Lemma 4.9 in
			(4.36) with p≥1 such that
			N/2(1-1/p)+1/2<1 (i.e.,
			p∈[1,N/N-1)) we deduce that 
    ∇
    			u_λ (t+τ)_p≤ C_pu_0_1t^--1/2,     ∀
    			t∈ [0,1]
 and therefore
			
    ∇ u_λ (2τ)_p≤ C(p,τ)u_0_1,    ∀λ >0.

			It can easily be shown that (4.41) implies (4.29). Therefore, (4.28) holds for all p∈[1,N/N-1). On the other hand, from (<ref>) we deduce that
			
    ∇ u_λ (t+τ)_p≤ C(p,τ),     ∀ t≥ 0,
    				∀λ >0,∀τ>0,
    				∀ p∈[1,N/N-1).

			
			To prove (<ref>) for all p∈ [1,∞] we use an iterative argument. Taking L^p norms in (<ref>) we obtain
			
    ∇_λ(t+τ)_p
    					≤ C_p t^--1/2u_0_1 
    +|a|λ∫_0^t ∇ G(t-s)_r F'(λ^-N
    					u_λ(s+τ))_∞∇ u_λ (s+τ)_q ds

			with r, q such that 1/r +1/q = 1+1/p.
			By virtue of (<ref>) we can choose q in the interval
			[1,N/N-1). The parameter r will also be chosen in [1,N/N-1) for guaranteeing the integrability of ∇ G(t-s)_r in the interval [0,t]. In this way we will obtain all thep∈[N/N-1,N/N-2) . In these ranges of
			r, q and p, from (<ref>) we obtain
			
    ∇_λ(t+τ)_p
    				   ≤ C_p t^--1/2u_0_1
           +|a| C(q,τ)∫_0^1
    				(t-s)^--1/2 ds 
       ≤ C{t^--1/2 + t^-+1/2}.

			From (<ref>) we deduce 
			
    ∇ u_λ (2τ)_p ≤ C(p,τ),
    			    ∀ p∈[N/N-1,N/N-2)

			which implies (<ref>) for
			∈[N/N-1,N/N-2).
			
			Iterating this argument. we obtain (<ref>) for all p∈[1,∞].
		
		
		

CHAPTER: WEAKLY NON-LINEAR ASYMPTOTIC BEHAVIOR

		
		

§ INTRODUCTION

		
		In this chapter, we address the asymptotic behavior of weakly non-linear heat equations such as
		
    u_t-Δ u   = a·∇(F(u))
    		    in ^N×(0,∞)
    
    		u(0)   =u_0

		
	The problem consists in proving, under suitable conditions on the nonlinearity F, that
		
    t^u(t)-MG(t)_p→ 0    when     t→∞

		G being the fundamental solution to the linear heat equation
		
    G(x,t)=(4π t)^-N/2exp(-|x|^2/4t)

		and M the mass of the solution: M=∫ u_0(x) dx. In other words, one seeks to prove that  
		when  t→∞ the solutions to
		(<ref>), as a first approximation behave like the solutions to the
		linear heat equation.
		
		As we mentioned in the introduction, this phenomenon occurs if
		
    F(s)=|s|^q-1s

		with q>1+1/N. IIn the general case of nonhomogeneous nonlinearity, this phenomenon is expected to occur when  F behaves like |s|^q-1s with q>1+1/N in a neighbourhood of 
		s=0. In the next section we will prove that (<ref>) holds true if
		
    lim_|s|→0F(s)/|s|^(N+1)/N=0.

		
		These results are optimal when F(s)=|s|^1/N s, as we see in the next chapter, (<ref>) is not fulfilled since the solutions to (<ref>) in that case behave since self-similar solutions of (<ref>).
		
		In Section 5.3 we will study the case of the initial data in L^1(^N; 1+|x|) ∩ L^∞(^N). In this case we can estimate the convergence speed in (<ref>). In Chapter I we saw that this is possible for the linear equation.
		
		In <cit.> we obtain the second term in the asymptotic behavior of
		the solutions to (<ref>)  when  F(s)=|s|^q-1s and q>1+1/N. 
		It can be shown that the second term depends on q in a way that it's necessary to distinguish the cases q>1+2/N, q=1+2/N and
		1/N<q<1+2/N.
		
		

§ INITIAL DATA IN L^1(^N)

		
		The main result of this section is the following.		
		
			Suppose  that the nonlinearity F∈ C^1() satisfies
			
    lim_|s|→0F(s)/|s|^(N+1)/N=0.

			
			Then, for all u_0∈ L^1(^N) with M=∫u_0(x) dx, the solution to (<ref>) satisfies (<ref>) for all p∈[1,∞].
		
		
		
			By Theorem 4.3 in Chapter 4 we know  that u(1)∈ L^1(^N)∩
			L^∞(^N). Then u satisfies the integral equation
			
    u(t+1)=G(t)*u(1)
    			+∫_0^t a·∇ G(t-s)*F(u(s+1)) ds.

			
From Theorem 1.1 of Chapter 1 we know that by conservation of the mass, ∫u(x,1) dx=M, one has
			
    t^G(t)*u(1)-MG(t)_p → 0,     when     t→∞

			for all p∈[1,∞]. On the other hand
			
    t^G(t+1)-G(t)_p→ 0,    when    t→∞

			so it remains to prove that
			
    t^∫_0^t a·∇
    			G(t-s)*F(u(s+1)) ds_p → 0.

			
			Let
			
    ϵ(s)=F(s)/|s|^(N+1)/N.

			
			From (<ref>) we know that
			
    ϵ(s)→ 0,     when   |s|→0

			
			It holds
			
    ∫_0^t a·∇ G(t-s)
    					*F(u(s+1)) ds
       ≤∫_0^t/2a·∇ G(t-s)_p
    					F(u(s+1))_1 ds
           +∫_t/2^t a-∇ G(t-s)_1
    					F(u(s+1))_p ds.

			
			By means of an explicit computation it can easily be shown that
			
    ∇ G(t)_p≤ C_p t^--1/2.

			
			On the other hand
			
    F(u(s))_p≤ϵ(u(s))_∞u(s)_p(N+1)/N^(N+1)/N.

			
			therefore, using the estimate
			
    u(t)_p≤ C_p t^-u_0_1

			shown in Theorem 4.3 of Chapter 4 we deduce that
			
    ∫_t/2^t a·∇ G(t-s)_1
    					F(u(s+1))_p
    ≤ |a| C∫_t/2^t (t-s)^-1/2ϵ(u(s+1))_∞ (s+1)^-(p(N+1)-N)/2p ds
    ≤ 2|a| C(t2)^-(p(N+1)-N)/2p(t2)^1/2sup_s≥t/2+1ϵ(u(s))_∞
    
    					=C'sup_s≥t/2+1ϵ(u(s))_∞
    					t^-

			and since
			
    lim_t→∞sup_s≥t/2+1ϵ(u(s))_∞=0

			We see that
			
    t^∫_t/2^t a·∇ G(t-s)_1
    			F(u(s+1))_p ds→ 0,    when     t→∞

			
			On the other hand
			
    ∫_0^t/2a·∇ G(t-s)_p
    					F(u(s+1))_1 ds
       ≤|a| C∫_0^t/2 (t-s)^--1/2ϵ(u(s+1))_∞
    					(s+1)^-1/2 ds
       ≤|a| C(t2)^--1/2∫_0^t/2ϵ(u(s+1))_∞(s+1)^-1/2 ds.

			
			therefore it is enough to prove that
			
    t^-1/2∫_0^t/2ϵ(u(s+1))_∞
    			s^-1/2 ds→ 0    when    t→∞.

			
			Fix an arbitrary δ>0. Since ϵ(u(t))_∞→
			0  when  t→∞, there exists T>0 such that
			
    ϵ(u(s+1))_∞≤δ,    ∀ s≥ T

			and so
			
    t^-1/2∫_T^t/2ϵ(u(s+1))_∞ s^-1/2 ds≤ 2δ t^-1/2(t/2-T)^1/2≤√(2)δ.

			
			On the other hand
			
    t^-1/2∫_0^T
    			ϵ(u(s+1))_∞ s^-1/2 ds→ 0    when   
    			t→∞.

			
			This concludes the proof of (<ref>) and thus the proof of the theorem.
		
		
		
		This result was shown in <cit.> for initial data
			u_0∈ L^1(^N)∩ L^∞(^N).
		
		
		

§ INITIAL DATA IN L^1(^N; 1+|X|)∩ L^∞ (^N)

		
		In order to estimate the rate of convergence in (<ref>), as for linear heat equation, it must be assumed that the initial data has a certain decay at infinity. In the linear case it was enough to suppose u_0∈ L^1(^N;1+|x|). In the nonlinear case we need  u_0∈ L^1(^N; 1+|x|)∩ L^∞ (^N). On the other hand, it will be necessary to know the behavior of the non linearity in a neighbourhood of the origin.
		
		The following result holds.
		
			Suppose  that F∈ C^1() satisfies
			
    |F(s)|≤ c|s|^q,     ∀ s∈:|s|≤ 1

			with q>1+1/N. Let u_0∈ L^1(^N; 1+|x|)∩ L^∞
			(^N) be such that ∫u_0(x) dx=M.
			
			Then
			
    u(t)-MG(t)_p≤ C_p t^- p(t),    ∀ t≥ 1

			with
			
    p(t)=
    			-t^1/2    q>1+2/N
    
    			t^-1/2log (t+2)   q=1+2/N
    
    			t^-(N(q-1)-1)/2   1+1/N<q<1+2/N.
   
		
		
		
			The estimate of p is dependent on q>1+2/N,
			q=1+2/N, ó 1+1/N<q<1+2/N.
			In <cit.> it is shown that these estimates are optimal.
		
		
		
			The solution u to (<ref>) satisfies the integral equation 
			
    u(t)=G(t)*u_0 + ∫_0^t a·∇ G(t-s)*F(u(s)) ds

			
			As u_0∈ L^1(^N;1+|x|), by Lemma 1.2 of Chapter 1 we know that
			
    G(t)*u_0-MG(t)_p ≤ Cu_0_L^1(;|x|) t^--1/2.

			
			Thus it is enough to prove that			
			
    ∫_0^t a·∇ G(t-s)*F(u(s)) ds_p
    			≤ Ct^- p(t).

			
			We will proceed, like in the proof of Theorem 5.1, by decomposing the integral in the subintervals [0,t/2] and [t/2,t]. As u_0∈
			L^1(^N)∩ L^∞(^N) it holds
			
    u(t)_p≤ C_p(t+1)^-,    ∀ t>0

			for all p∈[1,∞]. In particular u∈
			L^∞(^N×(0,∞)).
			
			On the other hand, from (<ref>) we deduce  
			
    ∀ R>0, ∃ C_R>0: |F(s)|≤ C_R |s|^q,     ∀ s∈:
    			|s|≤ R.

			
			therefore, in the interval [t/2,t] it holds
			
    ∫_t/2^t a·∇ G(t-s)_1
    				F(u(s))_p ds
    				   ≤|a| C∫_t/2^t (t-s)^-1/2u(s)_pq^q ds
       ≤|a|C∫_t/2^t(t-s)^-1/2 (s+1)^-N(pq-1)/2p ds
       ≤|a|Ct^-N(pq-1)/2p t^1/2
       =|a|Ct^-t^-(N(q-1)-1)/2.

			
			It can easily be shown that 
			
    t^-(N(q-1)-1)/2≤ Cp(t)

			for all q>1+1/N and t≥ 1.
			
			In the interval [0,t/2] we have
			
    ∫^t/2_t a·∇ G(t-s)_p
    					F(u(s))_1 ds
    ≤|a| C∫^t/2_t (t-s)^--1/2u(s)_q^q ds
    ≤|a|C(t2)^--1/2∫_0^t/2
    					(s+1)^-(N/2)(q-1) ds.

			
			On the other hand,
			
    t^-1/2∫_0^t/2 (s+1)^-(N/2)(q-1) ds≤ Cp(t)

			for all q>1+1/N and t≥ 1. This proves the theorem.
		
		
		
			
				
  1.—
				In in case when
				
    |F(s)|≤ C|s|^q,    ∀ s∈

				Theorem 5.2 holds for initial data u_0∈
				L^1(^N; 1+|x|)∩ L^q (^N).
				
				
  2.—
				When q>1+2/N, we obtain the same convergence speed as in the linear heat equation case (see Lemma 1.2 of Chapter I) as p(t)=t^-1/2.
				
				
  3.—
				The results  can be generalized to any nonlinearity satisfying
				
    |F(s)|≤ C|s|^(N+1)/Nϵ(s)

				with ϵ(·) continuous and non-decreasing such that	 ϵ(s)→0. In this case we obtain (<ref>) with
				
    p(t)=max{ϵ((t/2+1)^-N/2),t^-1/2∫_0^t/2ϵ((s+1)^-N/2)
    				(s+1)^-1/2 ds}.

			
		
		
		

CHAPTER: SELF-SIMILAR ASYMPTOTIC BEHAVIOR

		
		

§ INTRODUCTION

		
In this chapter we address the self-similar asymptotic behavior of the system
		
    u_t-Δ u   =a·∇(F(u))
    		   en ^N×(0,∞)
    
    		u(0)   =u_0.

		It consists in finding a one-parameter family
		
    {u_M}_M∈={t^-N/2f_M(x√(t))
    		}_M∈

		of solutions to (<ref>) such that ∫ f_M(x) dx=M satisfies
		
    t^u(t)-u_M(t)_p→ 0     when     t→∞

		for p∈[1,∞].
		
		As we saw in the introduction, when the nonlinearity is homogeneous, F(s)=|s|^q-1s,
		self-similar solutions can only exist when		q=1+1/N.
		
		In this section we show that if q=1+1/N, the
		self-similar solutions exist and (<ref>) holds for the general solution
We will work with the self-similar variables introduced in Chapter III, in which the self-similar profiles become stationary solutions of a new convection-diffusion equation, and (<ref>) is equivalent to the convergence of the paths to these stationary states. The existence of stationary solutions (self-similar profiles) of the new equation and (<ref>) will be a consequence of an abstract result for dynamic systems with adequate properties of compactness and dissipativity. In order to ensure the compactness of the trajectories, we will work in the weighted Sobolev spaces presented in Chapter III. The dissipative property necessary to apply the aforementioned abstract result will be a consequence of the L^1(^N) contraction property which is satisfied by the semigroup associated to (<ref>).
		
		In the case of a more general nonlinearity we will see that
		
    lim_|s|→ 0F(s)/|s|^1/Ns=1

		the solutions to (<ref>) behave like the self-similar solutions of problem (<ref>) with homogeneous nonlinearity F(s)=|s|^1/Ns. 
		To demonstrate this results.we will use a stability theorem for perturbed dynamical systems due toV. Galaktionov and J. L. Vázquez
		<cit.>.
		
This chapter is organized as follows. In section 7.2 we will state the main results of this chapter.
In section 7.3, we will state and prove the abstract results on the existence of steady states and stability of dynamical systems. In section 7.4 we will obtain estimates for solutions in the weighted Sobolev spaces. In section 7.5 we will prove the main result. We will conclude this chapter by proving the asymptotic self-similar behavior for nonlinearities
which satisfy (<ref>).
		
		

§ STATEMENT OF THE MAIN RESULTS

		
		Let us consider the equation
		
    u_t-Δ u=a·∇ (|u|^1/Nu)  in ^N×(0,∞).

		As we saw in the introduction, if u=u(x,t) is a self-similar solution to (<ref>), i.e.,
		
    u(x,t)=t^-N/2f(x√(t))
 
		the profile f=f(y) has to satisfy the elliptic equation	
		
    -Δ f-y·∇ f/2 - N/2 f=a·∇
    			(|f|^1/Nf)  in ^N.

		The following result on existence of self-similar solutions holds.
		
			For every M∈ there exists a unique profile f_M∈ L^1()
			solution to (<ref>) such that
			
    ∫ f_M(y) dy=M.

			
			Moreover,
			
				
  (i)
				f_M∈ H^2(K)∩ C^∞(^N)∩ W^2,p(^N),     ∀
				p∈ [1,∞);
				
				
  (ii)
				f_M>0 (resp. <0) in ^N if M>0 (resp. M<0);
				
				
  (iii)
				f_M_1>f_M_2 in ^N if M_1>M_2.
			
		
		
		This result is shown in <cit.> by means of a fixed-point argument applied to the elliptic equation (<ref>). In this chapter we will not reproduce that proof, but that we will adopt the dynamical point of view of <cit.>.
		
		Let
		
    u_M(x,t)=t^-N/2f_M(x√(t))

		where f_M is the self-similar profile of Theorem 6.1.
		
		It can easily be shown that 
		
    u_M(0)=Mδ

		δ being the Dirac mass, i.e.,
		
    ∫ u_M(x,t)φ(x) dx→ Mφ(0),     when    
    		t→ 0^+

		for all φ∈(^N).
		
		Regarding the asymptotic behavior of the general solution to (<ref>), we have the following result:
		
			Let u_0∈ L^1(^N) be such that
			
    ∫ u_0(x) dx=M

			and let u=u(x,t) be the solution to (<ref>) such that u(0)=u_0.
			
			Then (<ref>) holds for all p∈[1,∞].
		
		
This theorem proves that when t →∞ the solutions to (<ref>) behave as the self-similar solutions. This theorem extends the results proved in Chapters I and II for the linear heat equation and for the Burgers equation in one spatial dimension.
		
		
		

§ STEADY STATES AND CONVERGENCE OF TRAJECTORIES FOR DYNAMICAL SYSTEMS

		
		Let (Z,d) be a complete metric space and
		{S(t)}_t≥ 0 a dynamical system Z that satisfies:
		
    (i)    S(t)∈ C(Z,Z),   ∀ t≥ 0;  
    (ii)    S(0)=I= identity;  
    (iii)    S(t+s)=S(t)∘ S(s),   ∀ t,s≥ 0;
    			
    (iv)    S(t)z∈ C([0,∞);Z),   ∀ z∈ Z.

		
		We have the following result of existence of steady states and convergence of trajectories
		
			Suppose  that ⋃_t≥ 0{S(t)z} is relatively compact in Z for all z∈ Z. Suppose  that there exist continuous functions ψ∈ C(Z;),ϕ∈ C(,) such that
			
    ψ(S(t)z)=ψ (z),    ∀ t≥ 0,∀ z∈ Z  
       ϕ(d(S(t)z_1,S(t)z_2))
    				 is strictly decreasing in t if z_1 z_2 and ψ(z_1)=ψ(z_2).

			
			Then for all M∈ R(ψ) (= range of
			ψ) there exists a unique  z_M∈ Z such that
			
    S(t)z_M=z_M,     ∀ t≥ 0  and ψ(z_M)=M

			and for all z∈ Z such that ψ(z)=M, it holds
			
    S(t)z→ z_M      when     t→∞.

		
		
		
			We proceed in several steps.
			
			
				
  Uniqueness.
				This is an immediate consequence of (<ref>). Indeed, if z_M∈ Z and z_M'∈
				Z satisfy
				
    S(t)z_M=z_M, S(t)z'_M=z'_M,     ∀ t≥ 0,
    				ψ (z_M)=ψ (z_M')=M

				then (d(S(t)z_M,S(t)z_M')) es independiente de
				t, and therefore, in view of (<ref>), z_M=z_M'.
				
				
  Existence of steady states and convergence of trajectories. 
		Fix arbitrary M∈ R(ψ) and z∈ Z  such that ψ(z)=M.
				
	Let us consider the
				ω-limit set ω(z) of the trajectory {S(t)z}_t≥ 0: 
				
    ω(z)={ η∈ Z:∃ t_n→∞,
    				S(t_n)z→η }.

				
				By the hypothesis of compactness of trajectories, we know that ω(z)≠ϕ. On the other hand,
				
    ω(z)⊂{ η∈ Z:ψ(η )=M }

				as ψ∈ C(Z;).
		
		Thus it is enough to prove that
				
    ω(z)⊂{ η∈ Z:S(t)η=η ,∀
    					t≥ 0 }.

				
				Indeed, from (<ref>) and (<ref>) we deduce that
				
    ω(z)⊂{ η∈ Z:S(t)η=η,∀ t≥
    				0,ψ(η)=M }

				and since ω(z)≠ϕ, this guarantees the existence of a solution z_M to (<ref>). Since, on the other hand, we know that this solution is unique, we deduce that
				
    ω(z)={z_M}

				which is equivalent to (<ref>).
				
	With the goal of obtaining (<ref>) we will prove that given any τ>0
				
    S(τ)η=η,    ∀η∈ω(z).

				Fix an arbitrary τ>0 and η∈ω(z). Then there exists a
				sequence t_n→∞ such that
				
    S(t_n)z→η

				and therefore, since S(t)∈ C(Z;) for all t>0, by the semigroup property we deduce that
				
    S(t+t_n)z→ S(t)η,    ∀ t>0.

				On the other hand, by (<ref>), ψ(S(τ)η)=ψ(η), and in view of (<ref>), the following alternative holds: either (<ref>) holds or 
				
    ϕ(d(S(t+τ)η ,S(t)η)) is strictly decreasing.

				Therefore, it is enough to prove that (<ref>) is not fulfilled.
				
				As ϕ∈ C(;), por (<ref>) it holds
				
    ϕ(d(S(t+τ)η ,S(t)η))=lim_n→∞φ_τ(t+t_n).

				where
				
    φ_τ(t)=(d(S(t+τ)z,S(t)z)).

				
				From (<ref>) we deduce  that φ_τ(·) is decreasing. As the
				trajectory ⋃_t≥ 0{S(t)z} is relatively compact in Z
				and ϕ∈ C(;) it follows that the limit
				
    lim_t→∞φ_τ(t)=φ_τ,∞

				exists. Combining (<ref>) and (<ref>) we see that
				
    ϕ(d(S(t+τ)η,S(t)η))
    				=φ_τ,∞,    ∀ t>0
 and therefore (<ref>) does not hold. This completes the proof.
			
		
		
		

§ ESTIMATES IN THE WEIGHTED SOBOLEV SPACES

		
		Let us consider the solution u=u(x,t) to
		
    u_t-Δ u   =a·∇(|u|^1/Nu)   en
    			^N×(0,∞)
    
    		u(0)   =u_0
   
		with u_0∈ L^1(^N)∩ L^∞(^N).
		As seen in Theorems 4.1 and 4.3 of Chapter 4, we deduce that  u∈
		C([0,∞);L^1(^N))
		∩ L^∞(^N×(0,∞))
		and
		
    u(t)_p≤ C_p u_0_1 t^-,    ∀ t>0

		from where we deduce that
		
    u(t)_p≤ C'_p(t+1)^-,    ∀ t>0

		For all
		p∈[1,∞] with C'_p depending on u_0_1 and u_0_p.

		We define the corresponding function v=v(y,s) in the self-similar variables:
		
    v(y,s)=e^sN/2u(e^s/2y,e^s-1).

		
		It can easily be shown that  v=v(y,s) solves		
    v_s-Δ v-y·∇ v/2-N/2
    		v   =a·∇(|v|^1/Nv)   en ^N×(0,∞)
    
    		v(0)   =u_0.

		
		On the other hand, from (<ref>) we deduce that
		
    v∈ L^∞(0,∞ ;L^1(^N)∩ L^∞(^N)).

		
	In order to ensure the compactness of the trajectory {v(s)}_s≥ 0 that is needed to apply the Theorem 6.3, we will work in the weighted Sobolev spaces studied in the chapter 3.
		
		We have the following result.
		 
			Suppose  that u_0∈ L^2(K)∩ L^∞(^N). Then the
			solution v=v(y,s) of (<ref>) satisfies
			
    v   ∈ L^∞(0,∞ ;L^2(K))∩
    				C([0,∞);L^2(K)) 
    
    				v   ∈ L^∞(1,∞ ;H^1(K)) .


			Moreover, the trajectory {v(s)}_s≥ 0 is relatively compact  in L^2(K)∩ L^∞(^N).
		
		
		
			We first observe that by the biunivocal correspondence (<ref>) between the solutions to (<ref>) and
			(<ref>), the system (<ref>) admits a unique solution that is the given by
			(<ref>).
			
			As
			
    ∫ u(x,t) dx=∫ u_0(x) dx=M,     ∀ t>0

			in view of (19) it can easily be shown that 
			
    ∫ v(y,s) dy=M,    ∀ s>0.

			Thus the orthogonal projection of v(s) on the first eigenspace of the operator L in L^2(K) is independent of s and it holds
			
    v(y,s)=Mφ_1(y)+ṽ(y,s)

			with ṽ(s)∈φ_1^⊥ for all s>0, φ_1 being the first eigenfunction
			φ_1(y)=(4π)^-N/2 K^-1(y)=(4π)^-N/2exp(-|y|^2/4).
			
			By virtue of (<ref>) one may observe that (<ref>) is equivalent to
			
    ṽ(s)∈ L^∞(0,∞ ;L^2(K))∩
    			C([0,∞);L^2(K)).

			The following holds:
			
    ṽ_s+Lṽ-N/2ṽ   =a·∇(|v|^1/Nv)
    			   en ^N×(0,∞)
    ṽ(0)   =ũ_0

			with ũ_0=u_0-Mφ_1.
			Multiplying the equation (<ref>) by ṽK and integrating in ^N we obtain			
    1/2d/dsṽ(s)_K^2+∇ṽ(s)_K^2-N/2ṽ(s)_K^2   ≤|a|(N+1N) ∫|∇ v| |v|^1/N|ṽ|K
       ≤ |a|(N+1N)v_∞^1/N∫|∇(Mφ_1
    					+ṽ)| |ṽ|K.

			
			By Proposition 3.1 of Chapter 3 we deduce that
			
    ∇ w_K^2 ≥N+1/2w_K^2,
    			    ∀ w∈φ_1^⊥∩ H^1(K)

			and therefore
			
    ∇ w_K^2-N/2w_K^2≥1/N+1∇
    				w_K^2,      ∀ w∈φ_1^⊥∩ H^1(K).

			
			By (<ref>), (<ref>) and thanks to the fact that v∈
			L^∞(^N×(0,∞)) we obtain
			
    d/dsṽ(s)_K^2+2/N+1∇ṽ(s)_K^2≤ C_1{ṽ_K+∫ |∇ṽ| |ṽ|K}

			for C_1>0 sufficiently large . On the other hand
			
    ∫|∇ṽ||ṽ|K≤ϵ∇ṽ_K^2
    			+1/4ϵṽ_K^2

			for all ϵ>0 and therefore, eligiendo ϵ>0
			sufficiently small  we deduce that
			
    d/dsṽ(s)_K^2 +1/N+1∇ṽ(s)_K^2
    			   ≤ C_2{ṽ(s)_K+ṽ(s)_K^2}
       ≤ C_3{1+ṽ(s)_K^2}.

			
		From the inequality (<ref>) of Chapter III it follows that, for all
			ϵ>0 there exists C_ϵ>0 such that
			
    w_K^2≤ϵw_1,K^2 +C_ϵw_2^2,
    			    ∀ w∈ H^1(K).

			
		Therefore, choosing ϵ>0 sufficiently small and keeping into account that ṽ∈ L^∞(0,∞; L^2(^N))
			we obtain
			
    d/dsṽ(s)_K^2 +1/2(N+1)∇ṽ(s)_K^2≤ C_4

			and therefore
			
    d/dsṽ(s)_K^2+1/4ṽ(s)_K^2≤ C_4.

			
			From this differential inequality we obtain that ṽ∈
			L^∞(0,∞; L^2(K)) and therefore v∈
			L^∞(0,∞; L^2(K)). On the other hand, it can easily be checked that v∈ C([0,∞); L^2(K)). Thus, (<ref>) is proven.
			
	From (<ref>) and thanks to the regularizing effect of the semigroup associated to system (<ref>), we deduce  (<ref>).
			
			Indeed, S(s) being the semigroup generated by the operator
			L-N/2I, for all τ>0, the function v solution to (<ref>)
			satisfies the integral equation 
			
    v(s+τ)=S(s)v(τ)+∫_0^s
    				S(s-σ)[a·∇(|v(σ+τ)|^1/N
    				v(σ+τ)) ] dσ.

			
			As a consequence of the estimate (<ref>) from Chapter 3 for the semigroup S(·) it holds
			
    S(s)w_1,K≤ C(1+1/√(s))w_K,
    			    ∀ w∈ L^2(K)

			para C>0 sufficiently large. Taking  H^1(K) norms in (<ref>) and applying (<ref>) we obtain
			
    v(s+τ)_1,K
       ≤ C(1+1/√(s))v(τ)_K 
         +C|a|∫_0^s(1+1/√(s-τ))
    					∇(|v|^1/Nv)(σ+τ)_K dσ
       ≤ C(1+1/√(s))v(τ)_K
         +C|a|(N+1N) v_L^∞(^N×(0,s+τ))^1/N∫_0^s(1+1/√(s-τ))
    					v(σ+τ)_1,K dσ.

			As v∈ L^∞(^N×(0,∞)), we obtain
			
    v(s+τ)_1,K≤ C{(1+1/√(s))
    				v(τ)_K +∫_0^s(1+1/√(s-σ))
    				v(σ+τ)_1,K dσ}

			para C>0 sufficiently large.
			
			We now need the following Gronwall lemma.
			
		Let T>0, A, B≥0 and φ∈ C([0,T]) a non-negative function
				such that
				
    φ(t)≤ A(1+t^-1/2)+B∫_0^t
    				(1+(t-s)^-1/2)φ(s) ds.

				
		Then, there exists a constant C>0 that depends only on T and B such that
				
    φ(t)≤ CAt^-1/2,     ∀ t∈[0,T].

			
			
			
				The proof is analog to that of Lemma 4.2 in Chapter 4.
	Multiplying (<ref>) by t^1/2 we obtain
				
    t^1/2φ(t)   ≤ A(1+t^1/2)
    					+Bt^1/2∫_0^t(1+(t-s)^-1/2)
    					s^-1/2s^1/2φ(s) ds
       ≤ A(1+T^1/2)
    					+Bt^1/2∫_0^t(1+(t-s)^-1/2)
    					s^-1/2ψ(s) ds

				with ψ(t)=sup_s∈[0,t] s^1/2φ(s). As ψ is non-decreasing it follows that 
				
    t^1/2φ(t)≤ A'
    				+Bt^1/2ψ(t)∫_0^t(s^-1/2+(t-s)^-1/2s^-1/2)
    				 ds

				with A'=A(1+T^1/2). As
				
    ∫_0^t(s^-1/2+(t-s)^-1/2s^-1/2) ds
    				=2t^1/2+∫_0^1(1-s)^-1/2 ds=2t^1/2+α

				we obtain
				
    t^1/2φ(t)≤ A'+B(2t+α t^1/2)ψ(t).

				
				Fix t_0∈[0,T] such that B(2t_0+α t^1/2)=1/2.
				So, for all t∈[0,t_0] we deduce that
				
    t^1/2φ(t)≤ A'+1/2ψ(t),    ∀ t∈[0,t_0]

				and taking supremums in this inequality we observe that
				
    ψ(t)≤ 2A',    ∀ t∈[0,t_0].

				
				For t≥ t_0, choose δ≤t_0/2. Decomposing the last integral in (<ref>) in the segments [0,δ],
				[δ,t-δ] and [t-δ,t], using the fact that ψ is non-decreasing and taking supremums in (<ref>) we obtain
				
    ψ(t)   ≤ A'+B'(∫_0^δ(s^-1/2+(t-s)^-1/2s^-1/2) ds +∫_t-δ^t
    						(s^-1/2+(t-s)^-1/2s^-1/2) ds)ψ(t)
       +B'∫_δ^t-δ(s^-1/2+(t-s)^-1/2s^-1/2)ψ(s) ds.

				with B'=BT^1/2. Choosing δ>0 sufficiently small we observe that
				
    B'(∫_0^δ(s^-1/2+(t-s)^-1/2s^-1/2) ds)≤1/2,
    				    ∀ t∈[t_0,T]

				y since On the other hand
				
    s^-1/2+(t-s)^-1/2s^-1/2≤δ^-1/2(1+δ^-1/2),
    				    ∀ t∈[t_0,T],∀ s∈[δ,t-δ]

				we obtain
				
    ψ(t)≤ 2A'
    						+B'δ^-1/2(1+δ^-1/2)∫_0^tψ(s) ds
    ≤
    						A'+2A'B'δ^-1/2(1+δ^-1/2)t_0+B'δ(1+δ^-1/2)
    						∫_t_0^tψ(s) ds.

				
				Applying the classical Gronwall lemma we conclude that
				
    ψ(t)≤(2A'+2A'B'δ^-1/2(1+δ^-1/2)t_0)
    					exp(B'δ^-1/2(1+δ^-1/2)(t-t_0)).

				
				Combining (<ref>) and (<ref>) we see that
				
    ψ(t)≤ CA,    ∀ t∈[0,T]

				with C>0 depending on B and T. Clearly (<ref>) implies (<ref>).
			
			
			

  
*End of proof to Proposition 6.4:
			Applying Lemma 6.5 in the inequality (<ref>) we see that there exists C>0 such that
			
    v(s+τ)_1,K≤ Cs^-1/2v(τ)_K,    ∀τ>0,
    				∀ s∈[0,1].

			
			Taking s=1 and keeping into account that v∈ L^∞(0,∞;
			L^2(K)) we obtain
			
    v(1+τ)_1,K≤ C,    ∀τ> 0

			which implies (<ref>).
			
			From (<ref>) and by the compactness of the injection H^1(K)⊂ L^2(K) we deduce that {v(s)}_s≥0 is relatively compact  in L^2(K). From the continuity of the injection L^2(K)⊂ L^1(^N) we deduce in particular that {v(s)}_s≥0 is relatively compact  en
			L^1(^N).
			
			On the other hand, from the estimates shown in Theorems 4.3 and 4.4
			of Chapter 4 we deduce that
			
    v∈ L^∞(1,∞;W^1,p(^N))

			for all p∈[1,∞]. As W^1,p(^N)⊂ L^∞(^N)
			for p>N, by interpolation, we deduce that {v(s)}_s≥0 is relatively compact in L^∞(^N).
			
			Therefore, the trajectory is relatively compact  in L^2(K)∩
			L^∞(^N). This concludes the proof.
		
		
		
		
		

§ STRONG CONTRACTION PROPERTY IN L^1(^N).

		
		The following result holds.
		
		Let u_0, u_1∈ L^2(K)∩ L^∞(^N) be such that u_0 u_1 y
			
    ∫ u_0(y) dy=∫ u_1(y) dy.

			Let v and w be the corresponding solutions to (<ref>).
			
			Then v(s)-w(s)_1 is a strictly decreasing function of
			s.
		
		
		
			It is enough to show that (v(s)-w(s))^+_1 and
			(v(s)-w(s))^-_1 are strictly decreasing functions. As the proof is analog in both cases we will only consider the function 			
	
    θ(s)=(v(s)-w(s))^+_1=∫_Ω(s)(v(y,s)-w(y,s)) dy

			where
			
    Ω(s)={ y∈^N: v(y,s)>w(y,s) }.

			
			By Theorem 4.3 of Chapter 4 and taking into account (<ref>), which holds between the solutions to (<ref>) and (<ref>), we see that v∈ C
			((0,∞);W^2,p(^N)) for all p∈ (1,∞). In particular, v(s) and w(s) are continuous functions of y and therefore,
			Ω(s) is an open set for all s>0. Likewise
			
    Ω_*(s)={ y∈^N: v(y,s)<w(y,s) }

			is an open set for all s>0.
			
			Let us see that Ω(s) and Ω_*(s) are non-empty for all s>0. Indeed suppose that there exists s_0>0 such that Ω(s_0)=∅ (or
			Ω_*(s_0)=∅). As
			
    ∫ v(y,s) dy=∫ u_0(y) dy=∫ u_1(y) dy=∫ w(y,s) dy,
    			    ∀ s>0

			we deduce 
			
    v(y,s_0)=w(y,s_0),     ∀ y∈^N.

			The following lemma of backward uniqueness ensures that (<ref>)
			implies u_0=u_1, which goes against the hypotheses of Proposition 6.6.
			
			
			Let v and w be solutions to (<ref>) with initial data u_0 and u_1 respectively in L^2(K). Suppose  that there exists s_0>0 such that
				v(s_0)=w(s_0). Then, u_0=u_1.
			
			
			
		By (<ref>) we know that v,w∈ C([0,∞); L^2(K)). On another hand, integrating the inequality (<ref>) with respect to s and using
				(<ref>) we deduce  that v,w∈ L^2(0,s_0; H^1(K)).
				
				Set
				
    z(y,s)=v(y,s)-w(y,s)

				which satisfies
				
    z_s+Lz-N/2 z   =a·∇(b(y,s)z)    en
    					^N×[0,s_0]
    
    				z(s_0)   =0

				con
				
    b(y,s)=|v|^1/N v(y,s)-|w|^1/N w(y,s)/v(y,s)-w(y,s).

				
				As v,w∈ L^∞(^N×(0,∞)), we see that b∈ 
				L^∞(^N×[0,s_0]) therefore, since the gradient operator is continuous from L^2(K) in (H^1(K))^* (dual de
				H^1(K)) we see that
				
    a·∇(b(s)z(s))_(H^1(K))^*≤|a| Cb(s)z(s)_K
    				≤ |a|Cb(s)_∞z(s)_K.

				Theorem 1.1 on backward uniqueness J. M. Ghidaglia <cit.> (see also J.-L. Lions and B. Malgrage <cit.>), applied along with
				A=L-N/2I,  V=L^2(K), D(A)=H^1(K) y
				H=(H^1(K))^*, implies that z(s)=0 for all
				s∈[0,s_0] and therefore u_0=u_1. 
			
			
			

  
*End of proof of Proposition 6.6:
			Multiplying by (v(s)-w(s))^+ the equation satisfied by v-w and integrating in ^N we obtain
			
    θ'(s)   =∫_Ω(s)(v_s(y,s)-w_s(y,s)) dy
       =∫_Ω(s)Δ(v(y,s)-w(y,s)) dy
           +∫_Ω(s)[y·∇ v(y,s)/2+N/2
    			v(y,s)-y·∇ w(y,s)/2-N/2 w(y,s)] dy
           +∫_Ω(s) a·[∇(|v(y,s)|^1/N v(y,s)
    			)-∇(|w(y,s)|^1/N w(y,s))] dy.

			
			It holds
			
    ∫_Ω(s)[y·∇ v(y,s)/2+N/2
    					v(y,s)-y·∇ w(y,s)/2-N/2 w(y,s)] dy
    
    					=1/2∫_Ω(s)÷[y·(v(y,s)-w(y,s))
    					] dy
    
    					=1/2∫÷[y·(v(y,s)-w(y,s))^+] dy
    					=0

			put that y·(v(y,s)-w(y,s))^+∈ W^1,1(^N) as
			v(s),w(s)∈ H^1(K) for all s>0.
			
			Likewise, for the same reason
			
    ∫_Ω(s) a·∇(|v(y,s)|^1/N v(y,s)
    			-|w(y,s)|^1/N w(y,s)) dy=0.

			
			Therefore
			
    θ'(s)=∫_Ω(s)Δ(v(y,s)-w(y,s)) ds.

			
			Identity (<ref>) makes sense due to the regularizing effect of the semigroup associated with the equation (<ref>) we have
			v,w∈ C((0,∞);H^2(K))∩
			C^1((0,∞);L^2 (K)) and Thus 
			v,w∈ C((0,∞);W^2,1(^N))∩
			C^1((0,∞);L^1(^N)).
		
			From (<ref>) we deduce that θ non-increasing  as			
    ∫_{φ>0}Δφ≤0 dx,     ∀φ∈
    				W^2,1(^N)

			(see <cit.> for a detailed proof  of (<ref>)).  Note that if the set {φ>0} is regular, one would have
			∫_{φ>0}Δφ dx=∫_∂{φ>0}∂φ/∂νdσ≤0 as ∂φ/∂ν≤0 on the boundary {φ>0}. In the general case, we obtain (<ref>) thanks to the Sard lemma.
			
			Let us now see that θ is strictly decreasing. We argue by contradiction. If θ(s_0)=θ(s_1) for s_1>s_0, from
			(<ref>) and since θ is increasing we deduce that
			
    ∫_Ω(s)Δ(v(y,s)-w(y,s)) dy=0,
    				    ∀ s∈[s_0,s_1].

			
			We need the following lemma shown in <cit.>.
			
			
				Let φ∈ W^2,1(^N) be such that
				
    ∫_{φ>0}Δφ(x) dx=0.

				
				Then φ^+∈ W^2,1(^N) y
				
    Δφ^+=(φ^+) Δφ in ^N.

			
			
			In view of (<ref>) and applying Lemma 6.8 to the function φ=v(s)-w(s)
			we deduce that the function
			
    z(y,s)=
    				v(y,s)-w(y,s),   s∈ [s_0,s_1], y∈Ω(s)
    
    				0,   s∈ [s_0,s_1], y∉Ω(s)

			satisfies the equation
			
    z_s-Δ z-y·∇ z/2-N/2 z
    				=a·∇(b(y,s)z)  in ^N×(s_0,s_1)

			with
			
    b(y,s)=|v|^1/N v(y,s)-|w|^1/Nw(y,s)/v(y,s)-w(y,s),
    				   s∈ [s_0,s_1], y∈Ω(s)
    
    				0,   s∈ [s_0,s_1], y∉Ω(s)

			
			It holds
			
    b∈ L^∞(^N×(s_0,s_1))

			and
			
    z=0  in Ω=⋃_s∈(s_0,s_1)[Ω(s)×{s}]
    ={ (y,s)∈^N×(s_0,s_1):v(y,s)>w(y,s) }.

			
			As v,w∈ C(^N×(0,∞)) we see that Ω is an open subset of ^N×(s_0,s_1). On the other hand, we know that it is non-empty.
			
			Suppose that, using of a unique continuation argument, for certain
			s_2,s_3∈(s_0,s_1) with s_2<s_3, we prove
			
    z=0  in ^N×(s_2,s_3).

			Then, by definition of z it holds that Ω(s)=∅ For all
			s∈[s_2,s_3], which, like we have seen before, contradicts the hypothesis u_0 u_1.
			
When proving (<ref>) by means of a unique continuation argument, there is a technical difficulty that lies in the fact that the potential b ∈ L^∞ appears in the equation under a sign of derivation.
			
			To avoid this difficulty, we consider a ball B(y_0,ϵ) contained in ^N and s_2,s_3∈(s_0,s_1) such that s_2<s_3 and
			
    B(y_0,ϵ)×(s_2,s_3)⊂Ω.

			This is possible since Ω is a non-empty open subset of ^N+1.
			
			We define 
			
    n(y,s)=∫_0^(y_0-y)· a/|a|^2 z(y+λ
    				a,s) dλ.

			
			In (<ref>) the integral is taken on the segment I of ^N with a 
			y with
			y+(y_0-y)· a|a|^2)a, that is the projection of y on the hyperplane y· a=y_0· a of ^N that contains
			y_0 being orthogonal to the convection vector a.
			
		If y∈^N is an element of the cylinder
			
    C={ y∈^N:|y+a/|a|^2((y_0-y)·
    			a)-y_0
    			|<ϵ }

			then, Integrating the equation (<ref>) on the segment I we obtain
			that n=n(y,s) satisfies the equation
			
    n_s-Δ n-y·∇ n/2-(N-1)/2n=
    			-ba·∇ n  in  C×(s_2,s_3).

			On the other hand
			
    n=0  in {C∩ B(y_0,ϵ)}× (s_2,s_3),

			by unique continuation (cf. L. Hörmander
			<cit.>, Th. 8.9.1., p. 224), from (<ref>) and (<ref>) we deduce that
			
    n=0  in  C×(s_2,s_3)

			and thus 
			
    z(y,s)=-a·∇ n(y,s)=0  in  C×(s_2,s_3).

			
		Let H be the hyperplane of ^N, perpendicular to the vector a and which
			contains the origin. Given y∈^N, we write it as
			
    y=(y̅, (y· a)/|a|)

			where y̅∈^N-1 is the projection of y on the hyperplane H.
			
			We define 
			
    ρ(y̅,s)=∫_-∞^∞ z(y̅,λ,s) dλ=
    			∫_∞^∞ z(y+ta,s) dt.

			
			From (<ref>) we deduce that
			
    ρ(y̅,s)=0,    ∀(y̅,s)∈[H∩
    			B(0,ϵ)]×(s_2,s_3).

			
			On the other hand, integrating the equation (<ref>) in the direction of the vector
			a we deduce that
			
    ρ_s-Δ_y̅ρ-y̅·∇_y̅ρ/2
    			-(N-12)ρ=0  in  H×(s_2,s_3).

			(Δ_y̅ and ∇_y̅ denote the laplacian and the
			gradient in the variables y̅).
			
			By unique continuation we obtain
			
    ρ=0  in  H×(s_2,s_3)

			and since z≥ 0,
			
    z=0  in ^N×(s_2,s_3).

			
			This concludes the proof of  Proposition 6.6.
		
		
		

§ PROOF OF THE MAIN RESULTS

		
		In this part, we conclude the proofs of Theorems 6.1 and 6.2.
	To this end, we will apply Theorem 6.3.
		
		Let Z be the Banach space L^2(K)∩(^N), where (^N)
		is the Banach space of continuous and bounded functions endowed with the supremum norm. Let S(s) be the semigroup generated by the semilinear equation (<ref>).
		
		As we have seen before, the solutions v to (<ref>) with data
		inicial u_0∈ L^2(K)∩(^N) satisfy
		
    u∈ C([0,∞); L^2(K)∩(^N)).

		
		On the other hand,
		
    ∫ v(y,s) dy=∫ u_0(y) dy,    ∀ s≥ 0.

		
		Therefore the function
		
    ψ:     L^2(K)∩(^N)→
       φ→ψ(φ)=∫φ(y) dy

		satisfies hypothesis (6) of Theorem 6.3.
		
		On the other hand, by virtue of Proposition 6.6 we know that the function
		ϕ(·)=·_1 satisfies hypothesis (7) of Theorem 6.3.
		
		Applying Theorem 6.3 we obtain that ∀ M∈,
		
    (<ref>) admits a unique solution  f_M∈ L^2(K)∩(^N)  such that ∫ f_M(y) dy=M,

		
    ∀ u_0∈  L^2(K)∩(^N)  such that ∫ u_0(y) dy=M,  
    			 the solution v to (<ref>)
     satisfies 
    
    v(s)-f_M_L^2(K)∩ L^∞(^N)→0,
    			 when  s→∞.

		
		We conclude the proofs of Theorems 6.1 and 6.2 in several steps.
		
		

    Step  1.

		Regularity and positivity of the profiles f_M.
		
		The function
		
    u_M(x,t)=(t+1)^-N/2 f_M(x/√(t+1))

		is solution to (<ref>) with u_0=f_M. therefore, since f_M∈ L^1(^N),
		by Theorem 4.3 of Chapter 4 we deduce  that u(t)∈ W^2,p(^N)
		for all t>0 and p∈(1,∞), which implies that f_M∈
		W^2,p(^N) for all p∈(1,∞).
		
		On the other hand, since a·∇(|f_M|^1/N
		f_M)∈(H^1(K))^*, by elliptic regularity it holds f_M∈
		H^1(K). So, a·∇(|f_M|^1/N f_M)∈ L^2(K) and again, by elliptic regularity, f_M∈ H^2(K). From the continuity of the injection H^2(K)⊂ W^2,1 (^N) we deduce  that f_M∈
		W^2,1(^N).
		
		By the comparison principle from Theorem 4.3 we see that if u_0≥0, the solution u to (<ref>) satisfies u≥0 and therefore, the solution v of (<ref>) satisfies v≥ 0.
		
		If M≥ 0, take u_0∈ L^2(K)∩(^N) such that
		
    u_0≥ 0,  ∫ u_0(y) dy=M.

		As the corresponding solution v (<ref>) satisfies v≥0 and since v(s)→ f_M in L^∞(^N)  when  s→∞,
	we deduce that f_M≥ 0. In the same way we can show that
		M≤0, thus f_M≤0.
		
		By the Hopf maximum principle, we conclude that if M>0 (resp.		M<0) then f_M(x)>0 (resp. f_M(x)<0) for all x∈^N
		(cf. H. M. Protter and H. I. Weinberger <cit.>, Th. 5, p. 61).
		
		As the nonlinearity F(s)=|s|^1/Ns satisfies F∈
		C^∞(-{0}), by local elliptic regularity we deduce that
		f_M∈ C^∞(^N) for all M∈.
		
		

    Step  2.
 Uniqueness of the self-similar profiles.
		
		We have seen  that the self-similar profile es unique  in the clase L^2(K)∩(^N). Veamos that es unique  in L^1(^N). Suppose that there exist profiles f_M, f̃_M∈ L^1(^N) solutions to (<ref>) such that
		f_Mf̃_M and
		
    ∫ f_M(y) dy=∫f̃_M(y) dy=M.

		
		So,
		
    u_M(x,t)=(t+1)^-N/2 f_M((t+1)^-1/2 x);  ũ_M(x,t)=(t+1)^-N/2f̃_M((t+1)^-1/2 x)

	are solutions to (<ref>). Clearly
		
    ∫(u_M(x,t)-ũ_M(x,t))^+ dx=
    			∫(f_M(x)-f̃_M(x))^+ dx,    ∀ t>0.

		However the argument used in the proof of  Proposition 6.6 allows to prove that ∫(u_M(x,t)-ũ_M(x,t))^+ dx is a strictly decreasing function of t. This contradicts (<ref>),
		proves the uniqueness in L^1(^N) and concludes the proof of Theorem 6.1.
		
		

    Step  3.
 Monotonicity of the self-similar profiles.
		
		The proof of the uniqueness of the self-similar profiles done in the previous part allows to prove that two self-similar profiles cannot intersect. In the case M_1>M_2
		this implies that f_M_1>f_M_2 at any point.		
		

    Step  4.
 Asymptotic behavior for u_0∈
		L^2(K)∩(^N).
		
		From (<ref>) we deduce  that if u_0∈ L^2(K)∩(^N), the solution
		u=u(x,t)  to (<ref>) satisfies (<ref>) for all p∈ [1,∞].
		
		Given u_0∈ L^1(^N) such that
		
    ∫ u_0(x) dx=M

	we approximate it through a sequence {u_0,n}⊂ L^2(K)∩(^N) such that
		
    ∫ u_0,n(x) dx=M;      u_0,n→ u_0  in  L^1(^N)  when 
    		n→∞.

		
		Let u_n(x,t) be the solution to (<ref>) with initial data u_0,n. We know that
		
    u_n(t)-u_M(t)_1→ 0  when  t→∞

		with u_M being the self-similar solution of mass M. Since 
		
    u(t)-u_n(t)_1≤u_0-u_0,n_1,     ∀ t>0,∀
    		n∈

		we deduce 
		
    u(t)-u_M(t)_1→ 0  when  t→∞.

		
		Regarding the rescaled solutions u_λ(x,t)=λ^N
		u(λ x,λ^2 t), (<ref>) is equivalent to
		
    u_λ(·,1)→ f_M(·)  in  L^1(^N)  when 
    			λ→∞.

		
		On the other hand, from Theorems 4.3 and 4.4 of Chapter 4 we deduce that
		
    {u_λ(·,1)}_λ>0 is bounded in 
    			W^1,p(^N),    ∀ p∈[1,∞)

	and in particular
		
    u_λ(·,1)→ f_M(·)  in  L^p(^N), ∀
    			p∈[1,∞].

		
		As It can easily be shown , (<ref>) is equivalent to (<ref>). This concludes the proof of Theorem 6.2. 

§ MORE GENERAL NON-LINEARITIES

		
		The purpose of this section is to prove the following generalization of Theorem 6.2.
		
		
			Suppose that the non-linearity F∈ C^1() satisfies (<ref>).
			Then, for all u_0∈ L^1(^N) such that ∫ u_0(x) dx=M, la
			solution u=u(x,t) to (<ref>) satisfies (<ref>) For all
			p∈[1,∞], where u_M is the self-similar solution to (<ref>).
		
		
		
			Thanks to the estimate L^1-L^∞ of Theorem 4.3 in Chapter 4, it is enough to study the case where u_0∈
			L^1(^N)∩ L^∞ (^N). By  Theorem 4.3 of Chapter 4 we have
			
    u(t)_p≤ C_p(t+1)^-,    ∀ t >0

			for all p∈[1,∞].
			
			As we have not made any assumption on the behavior of the
			derivative of F, Theorem 4.4 of Chapter 4 is not applicable. 
			By the same method of proof we obtain
			
    u(t)_1-ϵ,p≤ C(ϵ,p)
    				t^--(1-ϵ)/2,     ∀ t>0

			for all ϵ>0, p∈[1,∞], with
			·_1-ϵ,p being the norm in W^1-ϵ,p(^N).
			
			The function v defined in (<ref>) satisfies in this case the following
			non-autonomous equation:			
    v_s-Δ v-y·∇ v/2-N/2v=
    			e^s(N+1)/2 a·∇(F(e^-Ns/2 v))    en ^N×(0,∞)
    
    			v(y,0)=u_0(y).

			Anew, (<ref>) is equivalent to proving that
			
    v(s)→ f_M  in  L^p(^N)  when  s→∞

			for all p∈[1,∞].
			
			The density argument used in the previous section shows that it's enough to consider initial data u_0∈ L^2(K)∩(^N).
			
			In view of (<ref>)–(<ref>) we know that
			
    v   ∈ L^∞(0,∞; L^p(^N)),  ∀ p∈[1,∞]
    
    			v   ∈ L^∞(1,∞;W^1-ϵ,p(^N)),  ∀
    			p∈[1,∞],∀ϵ>0.

			
			Using (<ref>), the condition (<ref>) imposed on the nonlinearity and by the method used in the proof  of Proposition 6.4 we obtain 
			
    v∈ L^∞(0,∞; L^2(K))
    
    			v∈ L^∞(1,∞;H^1-ϵ(K)),   ∀ϵ>0.
 
			and therefore the trajectory {v(s)}_s≥0 is relatively compact 
			in L^2(K).
			
			We will sue the following result on asymptotic behavior for perturbed dynamical systems, due to V. Galaktionov and J. L. Vázquez
			<cit.>.
			
			
		Let us consider a dynamical system in a Banach space X generated by the evolution equation
				
    w_t=A(w).

				and a perturbation
				
    w_t=B(t,w).

				
				Assume that the following properties hold:
				
					
  (a)
					The trajectories {w(t)}_t≥0 of (<ref>) are relatively compact in X. Moreover, given τ>0, if w^τ(t)=w(t+τ), the set {w^τ}_τ>0 is relatively compact in L_ loc^∞ (0,∞;X).
					
					
  (b)
					Given a solution w∈ C([0,∞);X) to (<ref>) and if t_j →∞ is such that w^t_j(·) converges to some function v ∈ L^∞_ loc(0, ∞; X), then  v is a solution to (<ref>).
					
					
  (c)
					The ω-limit set of the equation (<ref>) in X:
					
    {f∈ X |
    					.5∃ w∈ C([0,∞);X) solution to (<ref>)
    						and a sequence t_j→∞ such that w(t_j)→ f in X}

					is compact in X and uniformly stable in the following sense: For all  ϵ>0 there exists δ=δ(ϵ)>0 such that if w is a solution to (<ref>) with d(w(0),Ω)≤δ then
					d(w(0),Ω)≤ϵ, for all t>0.
				
				
				Then, under these assumptions, the ω-limit sets of the
				solutions w∈ C([0,∞);X) to (<ref>) are contained in
				Ω.
			
			
			With the goal of applying this result to the system (<ref>) we decompose
			v in the following way
			
    v(y,s)=f_M(y)+ṽ(y,s).

			As ∫ v(y,s) dy=∫ f_M(y) dy=M for all s≥0, ṽ(s)∈φ_1^⊥ for all s≥0. On the other hand, (<ref>) is equivalent to
			
    ṽ(s)→0  in  L^p(^N),  when  s→∞.

			The function ṽ is solution to the system:
			
    ṽ_2+Lṽ-N/2ṽ   =e^(N+1)s/2a·∇[F(e^-sN/2(f_M+ṽ(s)))]
           -a·∇(|f_M|^1/N f_M)   in   ^N×(0,∞)
    ṽ(0)   =u_0-f_M.
  
			
	We apply the Theorem 6.5 with X=L^2(K)∩φ_1^⊥. The system
			(<ref>) is (<ref>) and the non-perturbed system (<ref>) is
			
    ṽ_s+Lṽ-N/2ṽ=a·∇(|f_M+ṽ|^1/N
    					(f_M+ṽ))
    -a·∇(|f_M|^1/Nf_M)
    					 in ^N×(0,∞).

			
			We see that the three assumptions of Theorem 6.5 are satisfied:
			
				
  (a)
				From (<ref>) we deduce  that ṽ∈
				L^∞(1,∞;H^1-ϵ(K)). therefore the trajectories {ṽ(s)}_s≥0 of (<ref>) are relatively compact in L^2(K). By parabolic regularity, one can show that
				v_s∈ L^∞(1,∞; (H^ϵ(K))^*) for all ϵ>0. As ṽ∈ L^∞(1,∞;
				H^1-ϵ(K))∩
				W^1,∞(1,∞;(H^ϵ(K))^*), by Aubin-like compactness results (cf.<cit.>) we deduce  that the family of translations {ṽ^τ}_τ>0 is relatively compact in L_ loc^∞(0,∞;L^2(K)).
				
				
  (b)
				By virtue of the condition (<ref>) that the non-linearity satisfies, it can be easily be checked that if s_j→∞, then the translations {ṽ^s_j}_j converge in L_ loc^∞(0,∞; L^2(K))
				to a solution to the equation (<ref>).
				
				The ω-limit set in L^2(K)∩φ_1^⊥ reduces to
				Ω={0} because, due to  Theorem 6.2, all solution to (<ref>) with 
				initial data in L^2(K)∩φ_1^⊥ satisfies
				
    ṽ(s)→0  in  L^2(K)    when     s→∞.

				
				Let us see lastly the uniform stability of Ω={0}. By the L^1(^N) contraction property we see that if ṽ and w̃ are two solutions to (<ref>) with initial data ṽ_0 and w̃_0 it holds
				
    ṽ(s)-w̃(s)_1≤ṽ_0-w̃_0_1
    				≤ Cṽ_0-w̃_0_K,    ∀ s≥ 0.

				In particular, taking w̃_0=0 it holds
				
    ṽ(s)_1≤ṽ_0_1≤ Cṽ_0_K,
    					    ∀ s≥0.

				
				Using this estimates and techniques from the proof of Proposition 6.4 we obtain the stability of Ω={0}.
				
				As a consequence of Theorem 6.5 we deduce that the solutions to (<ref>)
				satisfy
				
    ṽ(s)→ 0  in  L^2(K)     when    s→∞.

				or, equivalently, the solutions (<ref>) satisfy
				
    v(s)→ f_M   in  L^2(K)    when     s→∞.

				
			Since on the other hand, {v(s)}_s≥0 is bounded in
				W^1-ϵ,p(^N) for all ϵ>0,  p∈[1,∞], one obtains (<ref>).
				
	This concludes the proof of Theorem 6.4.
			
		
		
		

CHAPTER: COMMENTS


		1.—The techniques from previous chapters 
		allows us to address more general equations of the form
		
    u_t-Δ u   =÷(F(u)) in ^N×(0,∞)
    
    		u(0)   =u_0∈ L^1(^N)
 
		with F=(F_1,… ,F_N)∈ C^1(;^N) such that F(0)=0.
		
		Let
		
    b_i   =lim_|s|→ 0F_i(s)/s
    y
    			a_i   =lim_|s|→ 0F_i(s)-b_is/|s|^1/Ns.

		The function
		
    v(x,t)=u(x-bt,t)

		with b=(b_1,…,b_N) satisfies
		
    v_t-Δ v   =÷(H(s))
    
    		v(0)   =u_0

		con
		
    H_i(s)= F_i(s)-b_is.

		By virtue of (<ref>), by the methods shown in Chapter 7, one can prove that the solutions to (<ref>) behave like the self-similar solutions
		to the equation
		
    v_t-Δ v=a·∇ (|v|^1/Nv)

		with a=(a_1,…,a_N).
		
		In this way, we obtain the following results .
		
		Let F∈ C^1(;^N) be such that los límites (<ref>) and (<ref>) exist.
			Let u_0∈ L^1(^N) be such that
			
    ∫ u_0(x) dx=M.

			Then the solution u=u(x,t) to (<ref>) satisfies
			
    t^u(t)-u_M(t)_p→ 0     when     t→∞

			for all p∈[1,∞], where
			
    u_M(x,t)=t^-N/2 f_M(t^-1/2(x+bt))

			and f_M is the self-similar profile of mass M to the equation (<ref>).
		
		
		Obviously if a=0, f_M is the profile of the heat kernel
		
    f_M=M(4π)^-N/2exp(-|x|^2/4).

		
		
		
		2.—The techniques presented in Chapter 7 allow for addressing systems of the form
		
    u_t-Δ u   =a·∇ (|u|^1/Nu)+g(x,t)
    
    		u(0)   =u_0.
  
		
		By means of a rescaling argument, it can be shown that this system admits self-similar solutions precisely if g is of the form
		
    g(x,t)=(t+1)^-N/2- 1h
    			((t+1)^-1/2x) 
    ∫ h(y) dy=0.

	(<ref>) assures the conservation of the mass of the solutions.
		
		In the usual self-similar variables, if g is of the form
		(<ref>) the system (<ref>) is converted into		
		
    v_s+Lv-N/2v   =a·∇(|v|^1/Nv)+h(y)
    
    		v(0)   =u_0.

		
	Ifi h∈ L^2(K)∩ L^∞(^N) and (<ref>) is satisfied (i.e., h∈φ_1^⊥), it is possible to think that there exists a one-parameter family of stationary solutions to (<ref>), i.e., of solutions of the elliptic system
		
    Lf_M-N/2 f_M   =a·∇
    		(|f_M|^1/Nf_M)+h  in ^N
    ∫ f_M(y) dy   =M

		and that, if u_0∈ L^2(K) with ∫ u_0(y) dy=M the solution to (<ref>) satisfy
		
    v(s) → f_M  in  L^2(K)     when    s→∞.

		
		The Theorems 7.1 and 7.2 of Chapter 7 provide affirmative answers to these questions for h=0.
		
		However, in the case h=0 an additional difficulty appears: the estimates of the trajectories in L^1(^N). Indeed, multiplying 
		in (<ref>) por (v) we obtain
		
    d/dsv(s)_1≤h_1

		and therefore
		
    v(s)_1≤u_0_1 +h_1s,    ∀ s≥ 0

		which does not imply that v∈ L^∞(0,∞;L^1(^N)).
		
		In <cit.> we have shown that if h is sufficiently small in
		L^1(^N;|y|), then the trajectory
		v of (<ref>) is bounded in L^1(^N). Usual arguments allow us to show that {v(s)}_s≥ 0 is relatively compact in L^2(K).
		
		Applying the techniques of Chapter 7 we obtain the following result.
		
		
		
			Let h∈ L^2(K)∩ L^∞(^N) be such that
			
    ∫ h(y) dy=0.

			There exists ϵ>0 such that if
			
    ∫ |h(y)||y| dy<ϵ

			the elliptic equation (<ref>) admits a unique solution f_M∈
			L^1(^N)∩ L^∞(^N) for every M∈.
			
			Moreover, for all u_0∈ L^1(^N) such that 
    ∫
    				u_0(x) dx=M
 the solution u=u(x,t) to
			
    u_t-Δ u   =a·∇
    			(|u|^1/Nu)+(t+1)^-N/2-1
    			h((t+1)^-1/2x)
    
    			u(0)   =u_0

			satisfies
			
    t^u(t)-u_M(t)_p→ 0     when     t→∞

			for all p∈[1,∞] where 
    u_M(x,t)=t^-N/2
    				f_M(t^-1/2x).

		
		
It should be noted that both in relation to the existence of self-similar solutions and in relation to asymptotic behavior, there are no restrictions on the mass M.
		
		The method used in the proof of this theorem, analogous to that developed in Chapter 7, will be applied immediately without restriction on the norm of h in L^1(^N;|x|) if we have an estimate in L^1(^N) for the solution to (<ref>) with initial data u_0=0. 
		
		
		3.—The extension of the results of Chapter 7 to  convection-diffusion equations with degenerate diffusion of the type
		
    u_t-Δ (|u|^m-1u)    =a·∇ (|u|^q-1u)  in ^N×(0,∞),
     or 
    
    			u_t-÷(|∇ u|^p-2∇ u)    =a·∇ (|u|^q-1u)   in ^N×(0,∞)

		is an open problem. 
		It would be particularly interesting to know if for these equations, the strong contraction property is satisfied in L^1(^N).
		
		
		4.—In the works of I. L. Chern and T. P. Liu <cit.> and S. Kawashima <cit.> 
the asymptotic behavior of parabolic convection-diffusion systems and mixed systems of parabolic-hyperbolic type is described in one spatial dimension.

It would be interesting to extend these results to higher spatial dimensions using the self-similar solutions built in Chapter 7.		
		
		5.—Let Ω be a regular domain of ^N. Let F∈
		C^1(,^N) be such that F(0)=0. Let us consider the evolution equation
		
    u_t-Δ u=÷(F(u))  in Ω×(0,∞).

	
		Integrating formally this equation on Ω it is observed that for the mass of the solutions to be preserved, it is necessary to impose the following boundary condition:
		
    ∂ u/∂ν=ν· F(u)  on ∂Ω×(0,∞)

		where ν is the exterior normal of the domain Ω.
		
This boundary condition ensures that the flow across the border is zero.
		
The system (<ref>)–(<ref>) can be completed the initial condition		

    u(0)=u_0∈ L^1(Ω).

		
It would be interesting to study existence and uniqueness of solutions for this problem, as well as its asymptotic behavior (in the article <cit.>, H.   Amann studies the existence and uniqueness of solutions for parabolic equations with nonlinear boundary conditions).
		
The asymptotic behavior may strongly depend on the domain Ω. In particular, whether or not it is bounded.
		
If Ω is bounded, the existence and uniqueness of a stationary solution for each mass can be expected. In other words,  one can expect that the system
		
    -Δ f_M=÷(F(f_M))    in Ω
       ∂ f_M/∂ν=ν· F(f_M)    on ∂Ω
       ∫_Ω f_M(x) dx=M

		has a unique solution for all M∈. 

It would be interesting to study the existence and uniqueness of these stationary solutions, and then, to address the asymptotic behavior of the solutions of the evolution problem.
		
When Ω is a cone of ^N, using the self-similar variables and defining
		
    v(u,s)=e^sN/2u(e^s/2y,e^s-1)

		We see that v satisfies
		
    v_s+Lv-N/2 v=e^-(N+1)s/2÷(F(e^sN/2v))
    		    in Ω×(0,∞)
       ∂ v/∂ν=-e^-(N+1)s/2(ν· F(e^sN/2v))
    		-(y·ν)/2v    on  ∂Ω×(0,∞)
       v(0)=u_0.

		
		The asymptotic behavior of the solutions will then depend on the properties of F. If F=(F_1,…,F_N) with F_1(s)=|s|^q_i-1s in an neighbourhood of the origin, q_i>N+1/N, one may expect that solutions v behave like those of the linear parabolic equation		
		
    v_s+Lv-N/2   =0   Ω×(0,∞)
    ∂ν/∂ν+(y·ν)v/2   =0   ∂Ω×(0,∞)
    
    		v(0)   =u_0.
		
		On the other hand, one might expect that  s→∞ the solutions to
		(<ref>) converge to a steady states, namely solutions to the elliptic problem
		
    Lf_M-N/2 f_M   =0   Ω
    ∂ f_M/∂ν+(y·ν)/2 f_M   =0   ∂Ω
    ∫_Ω f_M(x) dx   =M.
 
		
		In the case in that F_i(s)=a_1|s|^1/Ns in a neighbourhood of the origin, perhaps the solutions to (<ref>) behave like the autonomous system
		
    v_s+Lv-N/2v   =a·∇(|v|^1/Nv)    in
    			Ω×(0,∞)
    ∂ν/∂ν   =-(a·ν)|v|^1/Nv-(y·ν)/2 v
    		    on ∂Ω×(0,∞)
    
    		v(0)   =u_0

	which in turn will converge into the stationary solutions of the following elliptic equation
		
    Lf_M-N/2 f_M=a·∇(|f_M|^1/N f_M)      in Ω
    ∂ f_M/∂ν+(y·ν)/2 f_M=
    		-(a·ν)|f_M|^1/Nf_M      on ∂Ω
    ∫_Ω f_M(y) dy=M.
 

All the results stated in this section regarding the system (<ref>) - (<ref>) are nothing more than conjectures. They are therefore open problems.
		
		6.—The methods developed in the previous chapters allows also to address the asymptotic behavior of the solutions to 
convection-diffusion equations with initial data u_0 such that
		
    u_0(x)→ l     when     |x|→∞

		with l∈-{0}, under the condition that
		
    u_0-l∈ L^1(^N).

		
		Let us consider for example a homogeneous nonlinearity:
		
    u_t-Δ u   =a·∇(|u|^q-1u)
    
    		u(0)   =u_0.
 
		
	Assuming that u_0 satisfies (<ref>), we observe that
		
    v(x,t)=u(x,t)-l

		satisfies
		
    v_t-Δ v   =a·∇(g(v))    in
    			^N×(0,∞)
    
    		v(0)   =v_0

		with v_0=u_0-l and
		
    g(s)=|s+l|^q-1(s+l)-|l|^q-1l.

		
		If q≥1, then g∈ C^1(), and		
    g(s)=q|l|^q-1s+q(q-1)/2l^q-2s^2+o(s^2).

		
		The function
		
    w(x,t)=v(x-aq|l|^q-1t,t)

		satisfies
		
    w_t-Δ w   =a·∇(h(w))
    
    		w(0)   =v_0

		with
		
    h(s)=q(q-1)/2 l^q-2s^2+o(s^2).

		
		In dimensions N>1 it can be seen that
		
    lim_|s|→0h(s)/|s|^1/Ns=0.

		By virtue of the results from Chapter 6 we obtain 
		
    t^-w(t)-MG(t)_p→ 0     when     t→∞

		for all p∈[1,∞], where
		
    M=∫(u_0(x)-l) dx

		and G is the heat kernel.
		
		Thus, we obtain:
		
			If N>1, q≥1 and the initial data u_0 satisfies (<ref>) with l
			0 and (<ref>), the solution u=u(x,t) to (<ref>) satisfies
			
    t^u(·,t)-l-MG(·+aq|l|^q-1t,t)_p→ 0,  if 
    			t→∞

			for all p∈[1,∞].
		
		
		As
		
    lim_|s|→0h(s)/s^2=q(q-1)/2 l^q-2

	in dimension N=1, a different result holds  since, by virtue of the results of Chapter 6, we know that the solution of (<ref>) behaves like the self-similar solution
		
    w(x,t)=t^-1/2 f_M(t^-1/2x)

	of the equation
		
    w_t-w_xx=aq(q-1)/2 l^q-1(w^2)_x

		such that
		
    ∫_-∞^∞ f_M(x) dx=M.

		
		We may obtain the following result.
		
			If N=1, q≥1 and the initial data u_0 satisfies (<ref>) with l0
			and (<ref>), the solution u=u(x,t) to (<ref>) satisfies
			
    t^u(·,t)-l-t^-1/2
    			f_M(t^-1/2(·+aq|l|^q-1t)
    			)_p→0,     when     t→∞

			for all p∈[1,∞], where f_M is the self-similar profile with mass
			M of the Burgers equation (<ref>).
		
		
		7.—In <cit.> we have studied the self-similar solutions to convection-diffusion equations of the form
		
    u_t-Δ u+|u|^p-1u=a·∇(|u|^q-1u)  in ^N×(0,∞).

		
		The case a=0 was addressed in <cit.> and <cit.> where it's proven, in
		particular, that if 1<p<2/N, there exists a unique self-similar solution of the form
		
    u(x,t)=t^-1/p-1 f(t^-1/2x)

		with f∈ H^1(K),f>0.
In <cit.> it is likewise shown that there exist
multiple solutions in H^1(K) that change signs.

On the other hand, in <cit.> the existence of a family of self-similar solutions whose profiles satisfy
		
    lim_|x|→∞ |x|^2/(p-1)f_0(x)=b≥0

		is shown. It is a one-parameter family of  parameter b. 
In <cit.> it is shown that these solutions allow for proving the asymptotic behavior of the solutions to (<ref>) with a=0, while again assuming that the datum u_0 of u is such that the limit of |x|^2/(p-1)u_0(x)  when  |x|→∞ exists and is non-negative.
		
In <cit.> we have demonstrated the existence of self-similar solutions of the type (<ref>) for all a∈^N,1<p<1+2/N and
q=p+1/2. The uniqueness has been proven only for |a| sufficiently large.
		
It is important to note that the positive solution is unique in the class H^1(K) but not in general, because 
    u(x,t)=t^-1/p-1(1p-1)^1/(p-1).
 is a self-similar solution to (<ref>).
		
As we have previously indicated, in the case a = 0 there are radial and positive self-similar solutions of (<ref>) that tend to zero when
| x | →∞ but whose profile does not belong to H^1(K) (<cit.>).
		
		8.—

The functions of type (<ref>) are very singular solutions of the equation (<ref>). Indeed,  when 1<p<1+2/N, it can be seen that
		
    t^(1/p-1-N/2)∫_^N u(x,t)φ(x) dx=t^-N/2∫_^N f(t^-1/2x)φ(x) dy
    →(∫_^N f(x) dx)φ(0),    ∀φ∈(^N).

As 1/p-1>N/2, it can be seen that the solution u in the instant t=0, presents a singularity stronger than the Dirac mass, that is, than the source-type solutions.
The source-type solutions of (<ref>) are those which satisfy the initial condition
		
    u(0)=cδ

where c∈ and δ is the Dirac mass in the origin.
		
In <cit.> it is shown that if a=0 and 1<p<1+2/N the system
(<ref>), (<ref>) admits a unique solution. On the other hand, if a=0 and p≥1+2/N, H. Brezis and A. Friedman prove in the same work that there are no solutions to (<ref>), (<ref>).
		
The case a 0 was first considered by J. Aguirre and M. Escobedo <cit.>, who prove that if 1<p<1+2/N y
1<q<1+1/N, the problem (<ref>), (<ref>) admits a unique
solution.
		
N. Liu in <cit.> proves that if p≥ 1+2/N y
1<q≤ 1+p+1/2 there cannot exist solutions to (<ref>), (<ref>).
		
Regarding the very singular solutions, in <cit.> it's shown that if 1<p<1+2/N and 1≤ q≤ 1+p+1/2, the solution to (<ref>), (<ref>),  when  c→∞, goes to a very singular solution to  (<ref>). These results for a=0 were shown by S. Kamin and L. Peletier <cit.>.
		
		9.—In the introductory chapter of these notes, we mentioned that the asymptotic behavior of the solutions de
		
    u_t-Δ u=a·∇(|u|^q-1u)

		with 1<q< 1+1/N it is radically different from the case q>1+1/N.
		
		Let us first consider the case one spatial dimension N=1:
		
    u_t-u_xx+1/q(|u|^q-1u)_x   =0   in ×(0,∞)
    
    		u(x,0)   =u_0(x).
 
		As we mentioned, the solutions to (<ref>)  when  t→∞ behave like the entropy solutions of the hyperbolic equation:
		
    u_t+1/q(|u|^q-1u)_x   =0   in ×(0,∞)
    
    		u(x,0)   =Mδ
 
		whose uniqueness was shown by T. P. Liu and M. Pierre <cit.> and that are explicitly given as follows:	
		
    u_M(x,t)   =(xt)^1/(q-1)χ_(0,r(t))
    
    		r(t)   =cM^(q-1)/q t^1/q,   c=(qq-1)^(q-1)/q.

		
		In <cit.> the following result is shown:
				
			Let u_0∈ L^1() be such that M=∫ u_0. Then, for all 1≤
			p<∞ it holds
			
    t^1/q( 1-1/p)u(t)-u_M(t)_p→0    when     t→∞

			where u_M is the entropy solution of (<ref>).
		
		
From this results we deduce that the asymptotic behavior of the solutions to (<ref>) is described by the one-parameter family of fundamental entropy solutions of (<ref>). The diffusion term of the system (<ref>) therefore tends to disappear when t →∞. We will say that this type of behavior is strongly non-linear.
		
		Note that from (<ref>) and the structure of the solution u_M to (<ref>) we deduce that the solutions to (<ref>) satisfy:
		
    u(t)_∞≤ Ct^-1/q,    ∀ t>0

which, given that 1 <q <2, gives us faster decay  when t →∞ to that of the linear heat equation or when q ≥2.


In fact, the first difficulty that we must solve to prove Theorem 7.5 is obtaining an estimate of type (<ref>). We can obtain this estimate from the following entropy inequality:
		
			Let 1<q<2 and u=u(x,t) be a non-negative solution of (<ref>).
			Then it holds
			
    (u^q-1)_x≤1/t.

		
		
		
			Observe that the function w(u^q-1)_x satisfies the equation
			
    w_t-w_xx+[z-βw/z]w_x
    				+w^2^3+βw/z^2=0

			with z=u^q-1 and β=(2-q)/(q-1). It can be checked that 
			W(t)=1/t is a super-solution to the equation (<ref>). 
By comparison we deduce
			
    w(x,t)≤ W(t).

		
		
From the entropy inequality (<ref>) we deduce that if u is a non-negative solution to (<ref>) with M=∫ u_0, then
		
    0≤ u(x,t)≤(qM(q-1)t)^1/q.

Indeed, given arbitrary t>0 and x_0∈, thanks to  (<ref>), it holds that
		
    u^q-1(x,t)≥ B-((x_0-x)t) if x≤ x_0
 with B=u^q-1(x_0,t). Therefore,
		
    u^q-1(x,t)≥1/t(x-x_1)    if  0≤ x-x_1≤ Bt
 with x_1=x_0-Bt. 
	Integrating in this inequality with respect to x
		we obtain
		
    M=∫ u(x,t) dx≥∫_0^Bt(xt)^1/(q-1) dx
    		=(q-1q) u^q(x_0,t)t

		from where we deduce  (<ref>).
		
Having demonstrated (<ref>) for non-negative solutions, this estimate can be extended to any solution by the maximum principle.
		
The estimate (<ref>) allows to conclude the proof of Theorem 7.5 by means of a rescaling argument. We define 
		
    u_λ(x,t)=λ u(λ x,λ^qt)

		which satisfies
		
    u_λ,t-λ^q-2 u_λ,xx +u^q-1_λ
    		u_λ,x =0
       u_λ(x,0)=λ u_0(λ x)

		and we prove that
		
    u_λ(·,t)→ u_M(·,t)  in  L^p()     when λ→∞

		for all 1≤ p<∞ and all t>0 where u_M is the unique entropy solution of (<ref>).
	
These ideas allows to address the case of spatial dimensions  N≥ 2, granted additional technical difficulties.
		
Suppose that a=(0,…,0,1) and we consider the equation
		
    u_t-Δ u   =∂_N(|u|^q-1u)
    
    		u(x,0)   =u_0(x)

	with 1<q<1+1/N (henceforth ∂_N designates the partial derivative
		in the direction x_N) with a non-negative initial data.
		
In this case, the entropy inequality that we obtain for the non-negative solutions is:
		
    ∂_N(u^q-1)≤1/t

		from where it follows that
		
    u(t)_∞≤ Ct^-(N+1)/2q.

Again, the estimate (<ref>) gives a better decay than that of the solutions to the heat equation or  (<ref>) for q≥ 1+1/N.
		
After proving the estimate (<ref>) we introduce the scaling
	
    u_λ(x,t)=λ^α u(λ^1/2 x',λ^β
    		x_N,λ t)

		con x'=(x_1,…,x_N-1),α=(N+1)/2q y
		β=(1+N+q-Nq)/2q. The function u_λ satisfies
		
    u_λ,t-Δ' u_λ-∂_N(|u_λ|^q-1u)=
    		λ^1-2β∂_N^2 u_λ
       u_λ(x,0)=λ^α u_0(λ^1/2 x',λ^β x_N)

		where Δ' is the laplacian in the variables (x_1,…,x_N-1).
		
	It can be shown that, for t>0 fixed,
		
    u_λ(·,t)→ u_M(·,t)  in  L^p(^N)     when   t→∞

		for all 1≤ p<∞, where u_M(x', x_N,t) is the non-negative entropy solution of the equation
		
    u_t-Δ' u   =∂_N(|u|^q-1u)
    
    		u(x,0)   =Mδ
 
		
		In this way we can conclude that
		
    t^α(p-1)/pu(·,t)- u_M(·,t)_p      when λ→∞

		for all 1≤ p<∞.
		
The most notable difference with respect to the proof done in one spatial dimension is the need to prove the existence and uniqueness of the entropy solution for (<ref>). This result when N = 1 was proven in <cit.>. 
In <cit.> we prove the existence of a unique non-negative entropy solution to (<ref>). 
The uniqueness without any restriction on the sign of the solution is an open problem.

The entropy condition that we introduce in <cit.>, and whuch allows for characterizing the unique nonnegative fundamental solution is a variant
of the classical condition of S.   Kruzhkov <cit.> that takes into account the presence of the term of partial diffusion. 
More precisely, it's the following condition: for all non-negative test function ψ=ψ(x') ∈𝒟(^N-1) one must have
		
    ∂/∂ t|u-ψ(x')|-Δ'|u-ψ(x')|
       ≤∂_N[(|u|^q-1 u-|ψ(x')|^q-1ψ(x'))
    				(u-ψ(x'))]
    				+(u-ψ(x'))Δ'ψ(x').

Under this entropy condition, the uniqueness of the  entropy solutions with initial data in L^1(^N)∩ L^∞(^N) is shown in <cit.> by means of the Kruzhkov method <cit.>. 
The uniqueness of the fundamental non-negative solution requires additional arguments and is covered in <cit.>.
	
	
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