A note on one-dimensional time fractional ODEs
Yuanyuan Feng Lei Li Jian-Guo Liu Xiaoqian Xu
math.CA Primary 34A08




1]Yuanyuan Fengyuanyuaf@andrew.cmu.edu
2]Lei Lileili@math.duke.edu
3]Jian-Guo Liujliu@phy.duke.edu
4]Xiaoqian Xu xxu@math.cmu.edu
[1,4]Department of Mathematics, Carnegie Mellon University, Pittsburgh, PA 15213, USA
[2]Department of Mathematics, Duke University, Durham, NC 27708, USA.
[3] Departments of Mathematics and Physics, Duke University, Durham, NC 27708, USA.


A note on one-dimensional time fractional ODEs
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In this note, we prove or re-prove several important results regarding one dimensional time fractional ODEs following our previous work <cit.>. Here we use the definition of Caputo derivative proposed in <cit.> based on a convolution group.  In particular, we establish generalized comparison principles consistent with the new definition of Caputo derivatives. In addition, we establish the full asymptotic behaviors of the solutions for D_c^γu=Au^p. Lastly, we provide a simplified proof for the strict monotonicity and stability in initial values for the time fractional differential equations with weak assumptions.






§ INTRODUCTION

The fractional calculus in time has been used widely in physics and engineering for memory effect, viscoelasticity, porous media etc <cit.>. There is a huge amount of literature discussing time fractional differential equations. For instance, one can find some results in <cit.> using the classic Caputo derivatives.  In this paper, we study the following time fractional ODE:

    D_c^γu= f(t, u),   u(0)=u_0,

for γ∈ (0, 1)  and f measurable. Here D_c^γu is the generalized Caputo derivative introduced in <cit.>. As we will see later, this generalized definition is theoretically more convenient, since it allows us to take advantage of the underlying group structure.

As in <cit.>, we use the following distributions {g_β} as convolution kernels for β∈ (-1, 0):

    g_β(t)=1/Γ(1+β)D(θ(t)t^β).
 
Here θ(t) is the standard Heaviside step function, Γ(·) is the gamma function, and D means the distributional derivative on ℝ. Indeed, g_β can be defined for β∈ℝ (see <cit.>) so that {g_β: β∈ℝ} forms a convolution group. In particular, we have 

    g_β_1*g_β_2=g_β_1+β_2.

Here since the support of g_β_i (i=1,2) is bounded from left, the convolution is well-defined.  Now we are able to give the generalized definition of fractional derivatives:
 
Let 0<γ<1. Consider u∈ L_^1[0, T). Given u_0∈ℝ, we define the γ-th order generalized Caputo derivative of u, associated with initial value u_0, to be a distribution in 𝒟'(-∞, T) with support in [0, T), given by 

    D_c^γu=g_-γ*((u-u_0)θ(t)).

Iflim_t→ 0+1/t∫_0^t|u(s)-u_0|ds=0, we call D_c^γu the Caputo derivative of u.

As in <cit.>, if the function u is absolutely continuous, the generalized definition reduces to the classical definition. However, the generalized definition is theoretically useful because it reveals the underlying group structure (see Proposition <ref>).

Let T>0. A function u∈ L_^1[0, T) is a weak solution to (<ref>) on [0, T) with initial value u_0, if f(t, u(t))∈𝒟'(-∞,T) and the equality holds in the distributional sense. We call a weak solution u a strong solution if  (i). lim_t→ 0+1/t∫_0^t|u(s)-u_0|ds=0;  (ii). both D_c^γu and f(t, u(t)) are locally integrable on [0,T).

By the group property (<ref>), we have

Suppose f∈ L_^∞([0,∞)×ℝ; ℝ). Fix T>0. Then, u(t)∈ L_^1[0, T) with initial value u_0 is a strong solution of (<ref>) on (0, T) if and only if lim_t→ 0_+1/t∫_0^t  |u(s)-u_0|  ds=0 and it solves the following integral equation 

    u(t)=u_0+1/Γ(γ)∫_0^t(t-s)^γ-1f(s, u(s))ds, ∀ t∈ (0, T).



Using this integral formulation, the following has been shown in <cit.> 

Suppose f:  [0,∞)× (α, β)→ℝ is continuous and locally Lipschitz continuous in u. For any given initial value u_0∈ (α, β), there is a unique strong solution, which either exists globally on [0,∞) or approaches the boundary of (α, β) in finite time. Moreover, this solution is continuous on the interval of existence.


Below in Section <ref>, we will establish some generalized comparison principles consistent with the new definition of Caputo derivatives. In Section <ref>, we establish the full asymptotic behaviors of the solutions for D_c^γu=Au^p. In Section <ref>, we provide a new proof for the strict monotonicity and stability in initial values with weak assumptions.




§ GENERALIZED COMPARISON PRINCIPLES


The comparison principles are important in the analysis of time fractional PDEs (See <cit.>). There are many versions of comparison principles proved in literature using various definitions of Caputo derivatives. In <cit.>,  the authors assumed f(t,·) to be non-decreasing. In <cit.>, f(t,·) was assumed to be non-increasing.
In <cit.>, there is no assumption on the monotonicity of f(t,·), but the function v is assumed to be C^1 so that the pointwise value of D_c^γv can be defined. Combining these ideas and establishing a crucial lemma (Lemma <ref>), we prove some generalized comparison principles in this section.  Similar to <cit.>, we define the inequality in the distributional sense:

Let U be an open interval. We say f∈𝒟'(U) is a nonpositive (nonnegative) distribution if for any φ∈ C_c^∞(U) with φ≥0, we have  ⟨ f, φ⟩≤ 0   (⟨ f, φ⟩≥ 0). 
We say f_1≤ f_2 in the distributional sense for f_1, f_2∈𝒟'(U), if f_1-f_2 is nonpositive. We say f_1≥ f_2 in the distributional sense if f_1-f_2 is nonnegative.


In order to prove the comparison principle, we first prove the following auxiliary lemma:

Suppose u∈ L_^1[0, T) and lim_t→ 0^+1/t∫_0^t|u(s)-u_0| ds=0. If there exists a function  f∈ L_^1(0, T) such that on interval (0, T) we have in the distributional sense that
D_c^γu≤ f,
then for any given A∈ℝ, we have in the distributional sense

    D_c^γ(u-A)^+≤χ(u≥ A) f,  on  (0,T).



First, recall the following result in <cit.>: if u∈ C[0, T)∩ C^1(0, T) and u↦ E(u) is C^1 and convex, we have

    D_c^γE(u)≤ E'(u)D_c^γu.

Now let us consider η∈ C_c^∞(-1, 0) with η≥ 0 and ∫η dt=1. Define η^ϵ(t)=1/ϵη(t/ϵ) and u^ϵ=η^ϵ*u.
As showed in <cit.>,  u^ϵ(0)→ u_0 and u^ϵ(t)→ u(t) in L_^1[0,T).

Denote E(u)=(u-A)^+ and define  E^δ(u)=(E*η^δ)(u). Clearly, (E^δ)'(u)=η^δ*χ(u≥ A) is nonnegative and increasing,  which implies that E^δ is a convex increasing function. Then, we have

    D_c^γE^δ(u^ϵ)≤ (E^δ)'|_u^ϵD_c^γu^ϵ.

It is not hard to see  lim sup_ϵ→ 0(E^δ)'|_u^ϵD_c^γu^ϵ≤ (E^δ)'|_u f(t). Since E^δ(u^ϵ) converges to E^δ(u) in L_^1 and E^δ(u^ϵ(0)) converges to E^δ(u_0), according to Definition <ref>, D_c^γE^δ(u^ϵ)→ D_c^γE^δ(u) as distributions. Moreover,  notice that the inequality is preserved in the distributional sense (Definition <ref>). We have D_c^γE^δ(u)≤  (E^δ)'|_u f(t). Taking δ→ 0, similarly we have D_c^γE^δ(u) converges as distributions to D_c^γ(u-A)^+. Then the right hand side of (<ref>) converges to χ(u≥ A)f(t), and the inequality is preserved in the distributional sense.


As is well-known, if u∈ H^1(0, T), D(u-A)^+=χ(u-A) Du. Since Caputo derivative is nonlocal, the equality is no longer true in general. However, we have similar inequalities and Lemma <ref> provides an answer.


Suppose u(t) is a locally integrable function with lim_t→ 0^+1/t∫_0^t|u(s)-u_0| ds=0. Let A∈ℝ and t_1∈ (0, T) is a Lebesgue point. If u≤ A for a.e. t≤ t_1, and on the interval (t_1, T) we have D_c^γu≤ 0 in the distributional sense, then we have u≤ A, a.e. (0, T).

Let u^ϵ be the mollification in the proof of  Lemma <ref>. Consider v^ϵ=u^ϵ-C(ϵ) θ(t)/Γ(1+γ)t^γ such that v^ϵ≤ A for t∈ [0, t_1+ϵ]. C(ϵ)→ 0 since t_1 is a Lebesgue point. Applying Lemma <ref>, D_c^γ(v^ϵ-A)^+≤χ(t≥ t_1+ϵ)(D_c^γu^ϵ-C(ϵ))≤χ(t≥ t_1+ϵ)(D_c^γu^ϵ-η_ϵ*D_c^γu).  Taking ϵ→ 0 yields D_c^γ(u-A)^+≤ 0. The details are left to readers. Now several versions of comparison principles can be stated as follows:



  * Suppose u_i∈ L_^1[0, T) with lim_t→ 0^+1/t∫_0^t|u_i(s)-u_i,0| ds=0 (i=1,2). Suppose u_1(t)≤ u_2(t) on [0, t_1] for a Lebesgue point t_1, and  the γ-th Caputo derivatives of u_1, u_2 on [0, t_1] are locally integrable. Define

    h_i(t)=u_i,0+1/Γ(γ)∫_0^t∧ t_1(t-s)^γ-1D_c^γu_i(s) ds, i=1,2.

Then, h_1(t)≤ h_2(t) for all t∈ [0, T]. Moreover,  assume there exists a measurable function f(t,u) such that (i) f(·, u_i(·)) (i=1,2) is locally integrable on [t_1,T); (ii)  f(t,·) is non-decreasing on [t_1, T); (iii) D_c^γu_1≤ f(t, u_1) and D_c^γu_2≥ f(t, u_2) in the distributional sense on (t_1, T), then u_1≤ u_2 a.e. on [0, T).


  * Suppose u_i∈ L_^1[0, T) with lim_t→ 0^+1/t∫_0^t|u_i(s)-u_i,0| ds=0 (i=1,2). If u_1≤ u_2 on [0, t_1] for a Lebesgue point t_1 and D_c^γ(u_1-u_2)≤ f(t, u_1)-f(t, u_2) as distributions on (t_1, T), with f(t,·) being non-increasing on (t_1, T) and f(·, u_i(·)) (i=1,2) being locally integrable on [t_1, T), then u_1≤ u_2 a.e on [0, T).


  * Suppose u(t) is a continuous function on [0, T]. If u(t_1)=sup_0≤ s≤ t_1u(s) for some t_1∈ (0, T] and f(t)=D_c^γu(t) is a continuous function, then f(t_1)≥ 0.



(i). Clearly, D_c^γh_i=D_c^γu_i for t≤ t_1 and D_c^γh_i=0 for t>t_1. Let u=h_1-h_2, A=0 in Corollary <ref>, we find h_1≤ h_2.
On [t_1, T), we have

    u_1(t)≤ h_1(t)+1/Γ(γ)∫_t_1^t(t-s)^γ-1f(s, u_1) ds,     u_2(t)≥ h_2(t)+1/Γ(γ)∫_t_1^t(t-s)^γ-1f(s, u_2) ds.

As h_1(t)≤ h_2(t) and f(t,·) is non-decreasing, one has u_1(t)≤ u_2(t) (see <cit.>).

(ii). Apply Lemma <ref> for u_1-u_2 and A=0. (The proof is similar as in Corollary <ref>.)

(iii). Consider u^ϵ(t)=u(t)+ϵθ(t)/Γ(1+γ) t^γ, where ϵ>0. Then, t_1 is the unique maximizer of u^ϵ on [0,t_1]. Let f^ϵ=D_c^γu^ϵ=f+ϵ. It suffices to show

    f^ϵ(t_1)≥ 0, ∀ϵ>0.

Otherwise, there is an ϵ_0>0 such that f^ϵ_0(t_1)<0. Since f^ϵ_0 is continuous, we can find δ>0 such that on [t_1-δ, t_1] f^ϵ_0 is negative and u^ϵ_0(t)≤ u^ϵ_0(t_1-δ) for t≤ t_1-δ. Applying Corollary <ref>, we have u^ϵ_0(t)≤ u^ϵ_0(t_1-δ) for t∈ [t_1-δ, t_1], which is a contradiction. Taking ϵ→ 0 then gives the result.


Though the conditions here are weaker under the new definition of Caputo derivative, (ii) is essentially <cit.> and  (iii) is well-known for C^1 functions (see, for example <cit.>).



Now, we establish a generalized Grönwall inequality (or another version of comparison principle), consistent with the new definition of Caputo derivative. The main construction is inspired by <cit.>.

Suppose f(t,u) is continuous and locally Lipschitz in u. Let v(t) be a continuous function.
If D_c^γv≤ f(t, v) in the distributional sense, and D_c^γu=f(t, u), with v_0≤ u_0. Then, v≤ u on the common interval. Similarly, if we have D_c^γv≥ f(t, v) as distributions and v_0≥ u_0, then v≥ u on the common interval.



We only prove the first claim (the proof for the other is similar). By Proposition <ref>,  D_c^γu=f(t, u) with initial value u(0)=u_0 has a unique solution on the interval [0, T_b), where T_b is the largest time of existence. Moreover, u is continuous on [0, T_b). 


Fix T∈ (0, T_b). Pick M large enough so that u(t) and v(t) fall into [0, T]× [-M, M]. Let L be the Lipschitz constant of f(t,·) for the region [0, T]× [-2M, 2M]. Consider

    v^ϵ=v-ϵ w.

Here w=E_γ(2L t^γ) is the solution to D_c^γw=2Lw with initial value 1, where E_γ(z)=∑_n=0^∞z^n/Γ(nγ+1) is the Mittag-Leffler function <cit.>. Clearly, if ϵ is sufficiently small, v^ϵ falls into [0, T]× [-2M, 2M].  Then, we find that in the distributional sense

    D_c^γv^ϵ=D_c^γv-ϵ 2L w
    ≤ f(t,v)-ϵ 2Lw≤ f(t, v^ϵ)-ϵ Lw.

We claim that for all such small ϵ,

    v^ϵ(t)≤ u(t), ∀ t∈ [0, T].
 
 If not, define
 
    t_1=sup{t∈ (0, T]: v^ϵ(s)≤ u(s), ∀ s∈ [0, t] }.

 Since v^ϵ(0)=v_0-ϵ<u_0, by continuity  we have t_1>0. By assumption, (<ref>) is not true, and we have t_1<T.  Consequently, there exists δ_1>0, such that 
v^ϵ(t_1)=u(t_1)  and v^ϵ(t)>u(t) for t∈ (t_1, t_1+ δ_1).
 Moreover, 

    D_c^γ(v^ϵ-u)≤ f(t, v^ϵ)-ϵ Lw-f(t,u).

By continuity, for some δ_2∈ (0,δ_1), D_c^γ(v^ϵ-u) is a nonpositive distribution on the interval (t_1, t_1+δ_2). By Corollary <ref>, we have v^ϵ(t)≤ u(t) for t∈ (t_1, t_1+δ_2), which is a contradiction.  Hence, (<ref>) is true.
Taking ϵ→ 0 in (<ref>) yields the result on [0, T]. Since T is arbitrary, the result is true.




§ ASYMPTOTIC BEHAVIORS FOR A CLASS OF FRACTIONAL ODES

In this section, we study the solution curves to the following autonomous fractional ODEs:

    D_c^γu=Au^p,  u(0)=u_0>0.

The monotonicity of the solutions to (<ref>) and some partial results for the asymptotic behaviors have been established in our previous work <cit.>.  The asymptotic behaviors of the solutions for the A<0, p>0 case have also been discussed in <cit.>.
However, the discussion on all the range of A and p is not complete. Here, we will give a complete description on asymptotic behaviors of the solution curves.



By Proposition <ref>, the strong solution u to (<ref>) exists on [0, T_b) for T_b∈ (0,∞]. If T_b<∞, either lim_t→ T_b^-u(t)=0 or lim_t→ T_b^-u(t)=∞. We give a complete description regarding the solutions curves to (<ref>):

Consider (<ref>). If A=0, then u(t)=u_0. If A>0,  then all the solutions are strictly increasing on (0, T_b). If A<0, then all solutions are strictly decreasing before they touch 0. 


  * Suppose A>0. If p>1, then T_b<∞ and u(t)∼[Γ(pγ/p-1)/AΓ(γ/p-1)]^1/p-1(T_b-t)^-γ/p-1, as t→ T_b^-. If p=1, then u(t)=u_0E_γ(At^γ). If p<1, then there exist c_1>0 and c_2>0 such that c_1 t^γ/1-p≤ u(t)≤ c_2t^γ/1-p,   t≥ 1.

  * Suppose A<0. If p<0, the solution curve touches u=0 in finite time where the right hand side blows up. If p=0, then u=u_0+Ag_1+γ. If p>0, then T_b=∞, and there exist c_1>0, c_2>0 such that 
c_1t^-γ/p≤ u(t) ≤ c_2t^-γ/p,   t≥ 1.
 


The A=0 or p=0 cases are trivial. The monotonicity has been proved in <cit.>. The A>0, p>1 case has also been discussed there. Indeed, there is also an accurate estimate of T_b in  <cit.>. The p=1 case is trivial.  The A<0, p>0 case has been discussed in <cit.>. In fact, they established a version of comparison principle and used a subsolution and a supersolution to get c_1t^-γ/p≤ u(t) ≤ c_2t^-γ/p,    t≥ 1.
For the case A<0, p<0, since the solution is decreasing, we have D_c^γu≤ Au_0^p<0 before u touches zero. Hence, the claim follows.

Now, we establish the results for A>0, p<1 case. First, let us construct the sub-solution as follows:

    ω(t)=
    
    u_0,     t∈ [0,t_0],
    
    a t^γ/1-p,    t≥ t_0.

Here a>0 is to be determined and t_0 is determined by at_0^γ/1-p=u_0. Clearly, ω is absolutely continuous on any finite interval. For t<t_0, D_c^γω=0≤ Aω^p. For t≥ t_0, we have

    D_c^γω= aγ/(1-p)Γ(1-γ)∫_t_0^tτ^γ/1-p-1/(t-τ)^γdτ
    < a γ B(γ/1-p, 1-γ)/(1-p)Γ(1-γ)t^γ p/1-p =
    aΓ(γ/(1-p)+1)/Γ(γ p/(1-p)+1)t^pγ/1-p,

where B(·,·) is the Beta function. Clearly, if we choose a>0 such that aΓ(γ/(1-p)+1)/Γ(γ p/(1-p)+1)≤ Aa^p, then D_c^γω≤ Aω^p. Such a exists because p<1.

For the super-solution, let us consider

    v(t)=
    
    u_0+B_1t^γ/Γ(1+γ),    t∈ [0,1],
    
    B_2 t^γ/1-p,     t≥ 1.

B_2 is determined by B_2=u_0+B_1/Γ(1+γ). This choice of B_2 makes v absolutely continuous on any finite interval. We now determine B_1. On [0,1],  one has D_c^γv=B_1.
For t>1, we have

    D_c^γv=B_1γ/B(1+γ,1-γ)∫_0^1τ^γ-1/(t-τ)^γ dτ+
     B_2/Γ(1-γ)γ/1-p∫_1^tτ^γ/1-p-1/(t-τ)^γdτ.

On [1, 2], one has D_c^γv 
>B_1γ/B(1+γ,1-γ)∫_0^1τ^γ-1/(2-τ)^γ dτ=B_1 C_1(γ).
For t>2, we have

    D_c^γv>B_2 1/Γ(1-γ)γ/1-pt^γ p/1-p∫_1/t^1τ^γ/1-p-1/(1-τ)^γdτ
    ≥ B_2 t^γ p/1-p C_2(p,γ).

It is clear that there exists M_1(A, p, γ) such that as long
as B_2≥ M_1, D_c^γv≥ Av^p for t≥ 2 since p<1. For v to be a super-solution, one needs 

    u_0+B_11/Γ(1+γ)≥ M_1,     
    B_1min(1, C_1(γ))
    ≥ A max(u_0^p, (u_0+B_1/Γ(1+γ))^p 2^pγ/1-p).

Such B_1 exists since p<1. Hence, applying comparison principle Theorem <ref> yields the result.




§ STRICT MONOTONICITY AND STABILITY IN INITIAL VALUES


It is well-known that solution curves for well-behaved ODEs do not touch each other. However, for fractional ODEs, similar results are not trivial since the dynamics is non-Markovian. By the comparison principles (or generalized Grönwall inequality), if f(t,u) in (<ref>) is continuous and locally Lipschitz in u, u(0)<v(0) implies u(t)≤ v(t) for t≥ 0. However we do not have strict inequality. In <cit.>, the strict inequality has been established following a series of contraction techniques. Using our new definition of Caputo derivative, we provide a new proof of that solutions are strict monotone in initial values, by assuming f∈ L_^∞. 

The following lemma (a variant of <cit.> or <cit.>), is important:

Let r_λ(t)=-d/dtE_γ(-λΓ(γ)t^γ) be the resolvent for kernel λ t^γ-1 (in other words, r_λ(t)+λ∫_0^t(t-s)^γ-1r_λ(s)ds=λ t^γ-1).
Let T>0. Assume h∈ L^1[0,T], h>0 a.e., satisfying 

    h(t)-∫_0^tr_λ(t-s) h(s)ds>0,  a.e.,   ∀λ>0.


Suppose v∈ L^∞[0,T], then the integral equation 

    y(t)+∫_0^t(t-s)^γ-1v(s) y(s)ds=h(t)

has a unique solution  y(t)∈ L^1[0, T]. Moreover, y(t)>0, a.e..

The proof is exactly the same as <cit.>, though we only assume v∈ L^∞[0, T] here.
Next, we provide a new proof for the strict monotonicity in initial value. We also prove the stability of solutions with respect to initial values.

Assume that f(·,·)∈ L_^∞([0,∞)×ℝ). Moreover, assume for every compact set K, there is L_K>0 such that |f(t,u)-f(t,v)|≤ L_K|u-v| for a.e. (t,u), (t,v)∈ K.  Then, for a given initial value u_0,  the solution in L_^∞[0, T_b) is unique.  Further, we have


  * Any two solutions u_i∈ L_^∞[0, T_b^i) (i=1,2) with initial values u_1,0<u_2,0 satisfy u_1(t)<u_2(t) on [0, min(T_b^1, T_b^2)). 


  * For any T>0, M>0, there exists C(M,T)>0 such that any two solutions with u_i_L^∞[0, T]≤ M (i=1,2) and initial values u_1,0, u_2,0 satisfy
 
    u_1-u_2_L^∞[0,T]≤ C(M,T)|u_1,0-u_2,0|.

 


Fix T∈ (0, min(T_b^1, T_b^2)). There exists K compact such that for a.e t∈ [0, T], (t, u_i(t))∈ K. By Proposition <ref>, one has

    u_i(t)=u_i,0+1/Γ(γ)∫_0^t(t-s)^γ-1f(s, u_i(s)) ds.

The boundedness of f(s, u_i(s)) implies that u_i(t)∈ C[0, T]. If u_1,0=u_2,0, by taking the difference, |u_1(t)-u_2(t)|≤ C∫_0^t(t-s)^γ-1|u_1(s)-u_2(s)| ds and the uniqueness therefore follows.

 Now, assume u_1,0≠ u_2,0. Define y(t)=(u_2(t)-u_1(t))/(u_2,0-u_1,0), we  have

    y(t)+∫_0^t(t-s)^γ-1v(s)y(s) ds =1,  where v(s)=-1/Γ(γ)f(s,u_2(s))-f(s, u_1(s))/u_2(s)-u_1(s).

If u_1(s)=u_2(s), we define v(s)=0. Note that |v|≤ L_K/Γ(γ) a.e. for t∈ (0, T). By setting h=1 in Lemma  <ref>, one has

    1-∫_0^t r_λ(t-s) ds=E_γ(-λΓ(γ)t^γ)>0.

By Lemma <ref>, y(t)>0. Since y is continuous, satisfying 

    y(t)≤ 1+∫_0^t(t-s)^γ-1v_L^∞[0,T]y(s) ds,

we have y(t)≤ C(v_L^∞, T) by <cit.>. This verifies the last claim.




§ ACKNOWLEDGEMENTS

The work of J.-G Liu was partially supported by KI-Net NSF RNMS11-07444 and NSF DMS-1514826. Y. Feng was supported by NSF DMS-1252912.



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