Uniform in time $L^{\infty}$-estimates for nonlinear aggregation-diffusion equations
Jose A. Carrillo Jinhuan Wang
math.AP


Corresponding author: Jinhuan Wang

Uniform in time L^∞-estimates for nonlinear aggregation-diffusion equations
    Jose A. Carrillo^ 2, Jinhuan Wang^ 1
    December 30, 2023
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1-School of Mathematics, Liaoning University, Shenyang, 110036, P. R. China 

 email: wjh800415@163.com

 2-Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK.
email: carrillo@imperial.ac.uk






We derive uniform in time L^∞-bound for solutions to an aggregation-diffusion model with attractive-repulsive potentials or fully attractive potentials. We analyze two cases: either the repulsive nonlocal term dominates over the attractive part, or the diffusion term dominates over the fully attractive nonlocal part. When the repulsive part of the potential has a weaker singularity (2-n≤ B<A≤2), we use the classical approach by the Sobolev and Young inequalities together with differential iterative inequalities to prove that solutions have the uniform in time L^∞-bound. When the repulsive part of the potential has a stronger singularity (-n<B<2-n≤ A≤ 2), we show the uniform bounds by utilizing properties of fractional operators. We also show uniform bounds in the purely attractive case 2-n≤ A≤ 2 within the diffusion dominated regime.


Keywords: aggregation-diffusion equations, global in time uniform estimates, Stroock-Varopoulos inequality.



§ INTRODUCTION

In this paper, we analyse the Cauchy problem for the aggregation-diffusion equation

    {[                           ρ_t=Δρ^m + div(ρ∇ (U∗ρ)),              x∈ℝ^n, t> 0,;                                      ρ(x,0)=ρ_0(x),                    x∈ℝ^n, ].

where n≥ 2, the diffusion exponent m>1-2/n, and the initial data ρ_0(x)∈ L^1_+(ℝ^n)∩ L^∞(ℝ^n). The unknown function ρ(x,t) represents the density of individuals interacting pairwise via the potential

    U(x)=|x|^A/A-λ|x|^B/B,    2≥ A>B>-n  ,  λ≥0  .

The interaction potential U is attractive-repulsive for λ>0 and a fully attractive for λ=0. Here, we use the convention |x|^0/0:=log |x|. These models arise as generalizations of Keller-Segel models for chemotaxis, and they appear as macroscopic models of collective behavior, see <cit.> and the references therein. We will restrict to two different cases in this work that depending on the singularity of the potential we classify as: weakly singular when 2-n≤ B<A≤2 with λ≥ 0, and strongly singular when -n<B<2-n≤ A≤ 2 with λ>0.

There is an extensive literature for local-in-time well-posedness results in these models when the interaction potential U is chosen as the attractive Newtonian potential (λ=0, A=2-n), we refer to <cit.> and the references therein. Under the condition m>1-2/n, the existence of local-in-time weak solutions ρ(x,t)∈ L^∞((0,T),L^1_+(ℝ^n)∩ L^∞(ℝ^n)) to (<ref>) conserving the mass, i.e.,

    _ℝ^nρ(x) dx=_ℝ^nρ_0(x) dx=:M_0 ,

is ensured. This result was proven in <cit.> only for the attractive Newtonian potential. However, the generalization of their argument to the ranges of weakly singular interaction potentials treated below is not difficult. The case of strongly singular potentials is more cumbersome but the techniques developed in <cit.> should suffice. In this paper, we will assume the local-in-time well-posedness of these models, while focusing on the derivation of a-priori uniform in time estimates of the L^p-norms for the solutions for all 1≤ p≤∞ allowing to extend the local-in-time results to global-in-time weak solutions. We will develop a full well-posedness theory for a general class of aggregation-diffusion models elsewhere.

In our present case, only stationary and equilibrium solutions have been deeply studied in <cit.> for the fully attractive case λ=0 and the full range of exponents for A. The nonlinear diffusion term models very localized repulsion between individuals while the nonlocal potential takes into account non-local repulsion effects at short distances while attractive at long distances. The balance between repulsive and attractive terms determines if the density remains finite for all times uniformly or not. The exact balance in case λ=0 is given by m=1-An, called the fair-competition regime, as proven in <cit.>. In the fair-competition regime solutions may blow-up in finite time for large enough mass M. This is the case for the classical Keller-Segel model (n=2, m=1, A=0) exhibiting the celebrated critical mass phenomena as well as for the full fair-competition regime m=1-An with -n<A≤ 0. However, for the range of the parameters m>1-An, λ=0, and -n<A≤ 0, called the diffusion-dominated case, the existence of equilbrium solutions for all masses M>0 has been recently proven in <cit.>. We remind the reader that uniform in time estimates in L^∞ where proven in <cit.> for m>1 for the attractive Newtonian case in two dimensions (n=2, λ=0, A=0). We also refer to <cit.> where solutions are shown to converge towards a unique equilibrium profile for diffusion-dominated regimes in 2d.

In this work, we find regimes for the general aggregation-diffusion model (<ref>) with power-law potentials, in which there is a counterbalance due to repulsion (local or nonlocal) of the aggregation term given by the attractive A-term in U, leading to uniform L^∞-bounds. The asymptotic behavior in these cases is not clarified yet, since it could be given by equilibrium solutions, see the ranges covered in <cit.>, or all the solutions may convergence to zero, quite likely in cases where the attraction is not so strong and maybe overcome by the combined effect of the nonlocal repulsion and the nonlinear diffusion term. The asymptotic behavior will also be discussed elsewhere.

The strategy of proof for all the theorems in this work can be summarized as follows. The first step is that for any fixed p>1, we need to obtain estimates on the L^p-norm of the solution uniformly in time with the sharpest constants possible. In the second step, we derive an important iterative differential inequality with uniform bounds in the constants involved with respect to the L^p-exponent. Finally, the third step is to obtain the uniform L^∞-bound of the solution by utilizing a bootstrap iterative technique. This method follows a similar strategy to <cit.> but with different ingredients in each of the cases described below making them essentially different to the result in <cit.>. The next section treats the weak singularity case in which we take advantage of the repulsive nonlocal part to obtain the best estimates on the nonlocal term, the nonlinear diffusion term plays a less important role and it is used in order to get the regularization procedure. We also show in the purely attractive case, λ=0, that the nonlinear diffusion is able to overcome the nonlocal attraction in the diffusion-dominated case, i.e. for m>1-An. The final section deals with the strong singularity case in which we quantify the contribution of the repulsive nonlocal term by means of fractional Sobolev inequalities proved in <cit.>, and thus, we recover a similar result as in the weak singularity case.






§ UNIFORM L^∞-BOUND FOR THE WEAK SINGULARITY CASE

In this section, we prove a-priori uniform L^∞-bounds for solutions to the model (<ref>) in the attractive-repulsive case λ>0 with 2-n≤ B<A≤2 and the purely attractive case λ=0 with 2-n≤ A≤2. For simplicity, whenever λ>0, we take λ=1 in the proofs of this section. In fact, we subdivide for λ>0 in two subcases to discuss the properties: (i) the case 2-n< B<A≤2, (ii) the case 2-n= B<A≤2. For the subcase (i), we will make use of the Sobolev inequality and the Young inequality together with interpolation inequalities (Step 1) to deduce the differential iterative relation with uniform constants in the iteration exponents (Step 2). The uniform L^∞-bounds will be obtained as a consequence of these iterative inequalities (Step 3). In the subcase (ii), we can reduce to the subcase (i) via rearrangement inequalities. Finally for λ=0, the iterative inequalities need the additional use of the Hardy-Littlewood-Sobolev inequality.



 §.§ The case 2-n< B<A≤2 and λ>0

We can compute the Laplacian of the potential that in the distributional sense gives

    Δ U(x)=(A-2+n)|x|^A-2-(B-2+n)|x|^B-2=:f(|x|).

The presence of the repulsive part of the potential allows for the following fact that strongly prevents concentrations at the origin.


If 2-n<B<A≤2, and f(|x|) is defined in (<ref>), then there exists a zero point r_0>0 of f(|x|) such that

    f(|x|)<0,  0<|x|<r_0,
           f(|x|)≥ 0,  |x|≥ r_0.



From the definition of f(|x|) in (<ref>), we know that

    f(|x|)=|x|^B-2((A-2+n)|x|^A-B-(B-2+n)).

Due to 2-n<B<A≤2, and taking r_0:=(B-2+n/A-2+n)^1/A-B>0 satisfying f(r_0)=0, we can deduce
(<ref>) and (<ref>).

(Space-Time uniform L^∞-bound)
Assume that the non-negative initial data ρ_0∈ L^1(ℝ^n)∩ L^∞(ℝ^n), the diffusion exponent m>1-2/n, and 2-n<B<A≤2. If ρ is a global weak solution, then it satisfies a uniform in time L^∞-bound

    ρ_L^∞(ℝ_+; L^∞(ℝ^n))≤ C ,

for all λ>0.


As mentioned above, we follow the three-step procedure implemented in some related degenerated Keller-Segel models in <cit.>.

Step 1.- We first prove that the solution ρ has uniform in time L^p-bounds for any fixed p>1. Taking pρ^p-1 as a test function in the equation (<ref>), and integrating the result equation respect to x in ℝ^n, we deduce

    d/dt_ℝ^nρ^p dx
    =-4mp(p-1)(m+p-1)^2_ℝ^n|∇ρ^m+p-1/2|^2 dx +(p-1)_ℝ^nρ^pΔ (U*ρ) dx.

Now, we will focus on estimating the last term in the right-hand side. From (<ref>) and (<ref>), we estimate it as

    (p-1)_ℝ^nρ^pΔ (U*ρ) dx≤ (p-1)_ℝ^nρ^p(x)∫_|x-y|≥ r_0f(|x-y|)ρ(y) dy dx.

Noticing that 2-n<A≤ 2 implies |x-y|^A-2≤ r_0^A-2 for |x-y|≥ r_0 and B>2-n, then from (<ref>), we deduce that

    (p-1)_ℝ^nρ^pΔ (U*ρ) dx≤ (p-1)(A-2+n)r_0^A-2M_0_ℝ^nρ^p(x) dx,

where M_0 is defined by (<ref>).
Hence (<ref>) and (<ref>) gives that

    d/dt_ℝ^nρ^p dx
    ≤-2C_1_ℝ^n|∇ρ^m+p-1/2|^2 dx +C(A,B,n,M_0)(p-1)_ℝ^nρ^p(x) dx,

where 0<C_1<2mp(p-1)(m+p-1)^2 is a fixed constant independent of p.

Since p>1 and m>1-2/n, we have 1<p<(m+p-1)n/n-2. So, the interpolation inequality implies that

    ρ_L^p≤ρ_L^1^θ_1ρ^1-θ_1_L^(m+p-1)n/n-2,

where 1/p=θ_1+n-2/(m+p-1)n(1-θ_1), i.e.,

    θ_1=1-(p-1)(m+p-1)n/p((m+p-1)n-n+2),   1-θ_1=(p-1)(m+p-1)n/p((m+p-1)n-n+2).

Using (<ref>) and the Sobolev inequality, we get

    ρ^p_L^p≤ρ_L^1^pθ_1(S_n^-1/2∇ρ^m+p-1/2_L^2)^2p(1-θ_1)/m+p-1,

where S_n is the best constant of the Sobolev inequality
written as S_n h_L^2^*^2 ≤∇ h_L^2^2 for all h with finite L^2 gradient. Since m>1-2/n, we can directly compute that

    p(1-θ_1)/m+p-1= (p-1)n/n(m+p-1)-n+2<1,

which allows us to use the Young's inequality for (<ref>). Thus we get

    C(A,B,n,M_0)(p-1)ρ^p_L^p≤    σ_1 ∇ρ^m+p-1/2_L^2^2p(1-θ_1)/m+p-1q_2
       + C(σ_1)(C(A,B,n,M_0)(p-1))^q_1S_n^-q_1p(1-θ_1)/m+p-1ρ_L^1^q_1pθ_1

for any σ_1>0 and C(σ_1)=(σ_1 q_2)^-q_1/q_2q_1^-1, where q_1, q_2>1 satisfies 1q_1+1q_2=1.
Setting

    2p(1-θ_1)/m+p-1q_2=2     σ_1=C_1

into (<ref>) together with (<ref>), the differential inequality

    d/dt_ℝ^nρ^p dx
    ≤-C_1_ℝ^n|∇ρ^m+p-1/2|^2 dx +C(A,B,n,m,M_0,p)

holds.
Similar to (<ref>), we can also obtain

    ρ^p_L^p≤σ̅_1 ∇ρ^m+p-1/2_L^2^2p(1-θ_1)/m+p-1q_2+ C(σ̅_1)S_n^-q_1p(1-θ_1)/m+p-1ρ_L^1^q_1pθ_1

for any σ̅_1>0.
Hence (<ref>), with σ̅_1=C_1, and (<ref>) results in

    d/dt_ℝ^nρ^p dx
    ≤ -ρ^p_L^p +C(A,B,n,m,M_0,p).

A simple computation shows that

    ρ^p_L^p≤ρ_0^p_L^p+C(A,B,n,m,M_0,p),

i.e., for any fixed p>1, showing that the L^p-norm of ρ is uniformly bounded in time.

6pt

Step 2.- We now claim that the following iterative inequality

    d/dtρ_L^p_k^p_k≤-ρ^p_k_L^p_k +C̃ p_k^ℓ_2(ρ_L^p_k-1^p_k-1)^η

holds, where 0<η<2, 1<ℓ_2<n+1, p_k=2^k+1, k=0,1,2,..., and C̃ is a constant independent of p_k. The same procedure used to obtain (<ref>) in Step 1 gives that

    d/dtρ_L^p_k^p_k≤ -2C_1∇ρ^m+p_k-1/2^2_L^2 +C(A,B,n,M_0)p_kρ_L^p_k^p_k.

Using again the interpolation inequality and the Sobolev inequality, we infer that	

    ρ^p_k_L^p_k≤    ρ_L^p_k-1^p_kθ_2ρ^(1-θ_2)p_k_L^(m+p_k-1)n/n-2≤ρ_L^p_k-1^p_kθ_2(S_n^-1/2∇ρ^m+p_k-1/2_L^2)^2p_k(1-θ_2)/m+p_k-1,

where

    θ_2=1/p_k-n-2/n(m+p_k-1)/1/p_k-1-n-2/n(m+p_k-1),
       1-θ_2=1/p_k-1-1/p_k/1/p_k-1-n-2/n(m+p_k-1).

Hence, we easily compute that

    (1-θ_2)p_k/m+p_k-1<1,   m>1-2/n,  p_k=2^k+1.

Therefore, the Young inequality implies that

    C(A,B,n,M_0)p_kρ^p_k_L^p_k≤    C(σ_1)(C(A,B,n,M_0)p_k)^ℓ_2S_n^-ℓ_2p_k(1-θ_2)/m+p_k-1ρ_L^p_k-1^ℓ_2 p_kθ_2
       +σ_1 ∇ρ^m+p_k-1/2_L^2^2p_k(1-θ_2)/m+p_k-1ℓ_1

for any σ_1>0 with C(σ_1)=(σ_1 ℓ_1)^-ℓ_2/ℓ_1ℓ_2^-1, where ℓ_1, ℓ_2>1 satisfy 1ℓ_1+1ℓ_2=1. Setting

    2p_k(1-θ_2)/m+p_k-1ℓ_1=2     σ_1=C_1

into (<ref>) together with (<ref>), we infer that

    d/dtρ_L^p_k^p_k≤ -C_1∇ρ^m+p_k-1/2^2_L^2  +C(σ_1)(C(A,B,n,M_0)p_k)^ℓ_2 S_n^-ℓ_2p_k(1-θ_2)/m+p_k-1ρ_L^p_k-1^ℓ_2 p_kθ_2.


On the other hand, from (<ref>), we can get

    ρ^p_k_L^p_k≤ C(σ_1)S_n^-ℓ_2p_k(1-θ_2)/m+p_k-1ρ_L^p_k-1^ℓ_2 p_kθ_2+σ_1 ∇ρ^m+p_k-1/2_L^2^2,

where σ_1 and ℓ_2 are set in (<ref>). Thus, form (<ref>) and (<ref>) we deduce that

    d/dtρ_L^p_k^p_k≤-ρ_L^p_k^p_k+C(σ_1) (1+(C(A,B,n,M_0)p_k)^ℓ_2) S_n^-ℓ_2p_k(1-θ_2)/m+p_k-1(ρ^p_k-1_L^p_k-1)^ℓ_2 p_kθ_2/p_k-1.

Using (<ref>) and (<ref>), and after some tedious computations, we deduce that

    ℓ_2=m+p_k-1/m+p_k-1-(1-θ_2)p_k<n+1,  η:=ℓ_2 p_kθ_2/p_k-1=m-1+2/np_k/m-1+2/np_k-1<2,

and

    C(σ_1)(1+C(A,B,n,M_0)^ℓ_2) S_n^-ℓ_2p_k(1-θ_2)/m+p_k-1→ C(m,n,A,B,M_0),   p_k→∞,

i.e., there is a constant C̃:=C̃(m,n,A,B,M_0)>1 independent of p_k such that

    C(σ_1)(1+C(A,B,n,M_0)^ℓ_2) S_n^-ℓ_2p_k(1-θ_2)/m+p_k-1≤C̃.

Hence, (<ref>) implies that (<ref>) holds.

6pt

Step 3. In this step, we will use the iterative inequality (<ref>) to prove the L^∞-bound of ρ by induction. In (<ref>) from Step 1, we take p=p_0=2, then it holds that

    ρ^p_0_L^p_0≤ρ_0^p_0_L^p_0+C(A,B,n,m,M_0)≤C̅,

where C̅>1 is a constant independent of p_k and the time t.

Let y_k(t):=ρ^p_k_L^p_k, solving the iterative inequality (<ref>), we obtain

    (e^t y_k(t))'≤    C̃ p_k^ℓ_2y_k-1^ηe^t
    ≤    C̃ 2^n+12^(n+1)kmax{1,sup_t≥ 0y_k-1^2(t)}e^t,

where the last inequality used the facts p_k=2^k+1, η<2 and 1<ℓ_2≤ n+1.
Let a_k:=C̃ 2^n+12^(n+1)k>1 and D_0:=max{1,ρ_0_L^1,ρ_0_L^∞}, then there exists D>0 such that 
    D_0^p_k/2^k≤ D     k .

Then

    y_k(0):=ρ_0^p_k_L^p_k≤(max{1,ρ_0_L^1,ρ_0_L^∞})^p_k= D_0^p_k≤ D^2^k.

By (<ref>) and (<ref>), we have

    y_k(t)≤     a_kmax{1,sup_t≥ 0y_k-1^2(t)}(1-e^-t)+y_k(0)e^-t
    ≤     2 a_kmax{sup_t≥ 0y_k-1^2(t),D^2^k}.

From (<ref>) after some iterative steps, we have

    y_k(t)≤     2a_k(2a_k-1)^2(2a_k-2)^2^2···(2a_1)^2^k-1max{sup_t≥ 0y_0^2^k(t),D^2^k}
    
    =    (2^n+2C̃)^2^k-1 (2^n+1)^2·2^k-k-2max{sup_t≥ 0y_0^2^k(t),D^2^k},

Taking the power 1/p_k to above inequality, then

    ρ_L^p_k≤ 2^n+2 2^2(n+1)C̃max{sup_t≥ 0y_0(t),D}.

So, (<ref>) and (<ref>) imply that ρ_L^∞ is uniformly bounded in time.




 §.§ The case 2-n=B<A≤ 2 and λ>0

Notice that the potential -|x|^B/B is the Newtonian potential due to B=2-n. Hence we can formally compute

    Δ (U*ρ)(x)=    Δ(|x|^A/A*ρ)-nα_n((-Δ)ϕ*ρ)
    
    =    (A-2+n)|x|^A-2*ρ-nα_nρ(x).

Here α_n is the volume of n-dimensional unit ball and ϕ(x) is the fundamental solution of the Laplace equation.

(Space-Time uniform L^∞-bound)
 Assume that the initial data ρ_0∈ L^1_+∩ L^∞(ℝ^n), and the diffusion exponent m>1-2/n, and 2-n=B<A≤2. If ρ is a global weak solution, then it satisfies a uniform in time L^∞-bound

    ρ_L^∞(ℝ_+; L^∞(ℝ^n))≤ C ,

for all λ > 0.



We use the same strategy as proving Theorem <ref>. We will be able to reduce to the the same proof as in Theorem <ref> after the first step. We distinguish two cases.

6pt

Case A.- We start by considering the case A<2.

6pt

Step 1.- We first prove the uniform in time L^p-bound

    ρ^p_L^p≤ρ_0^p_L^p+C(A,B,n,m,M_0,p)  p>1.

For the case A<2, taking pρ^p-1, p>max{1,A-2+n/2-A}, as a test function in the equation (<ref>), integrating it and using (<ref>), we have

    d/dt_ℝ^nρ^p dx
    =   -4mp(p-1)(m+p-1)^2_ℝ^n|∇ρ^m+p-1/2|^2 dx -nα_n(p-1)_ℝ^nρ^p+1 dx
       +(p-1)(A-2+n)_ℝ^nρ^p(|x|^A-2*ρ) dx.

Notice that for any R>0, it holds that

    |x|^A-2≤ |x|^A-2χ_|x|<R(x)+R^A-2 ,

thus the integral coming from the attracting part can be controlled as

    _ℝ^nρ^p(|x|^A-2*ρ) dx   ≤_ℝ^nρ^p((|x|^A-2χ_|x|<R(x))*ρ) dx+_ℝ^nρ^p(R^A-2*ρ) dx
       =: I_R+M_0R^A-2_ℝ^nρ^p dx.

Since |x|^A-2 is locally integrable for A > 2-n, we can get by choosing R=R(A,n) small

    _ℝ^n|x|^A-2χ_|x|<R(x) dx≤nα_n/A-2+n,

which implies that |x|^A-2χ_|x|<R(x) is less concentrated than nα_n/A-2+nδ(x). Hence, we make use of the Riesz rearrangement inequality to obtain that

    I_R   ≤_ℝ^n(ρ^*)^p((|x|^A-2χ_|x|<R(x))*ρ^*) dx≤nα_n/A-2+n_ℝ^n(ρ^*)^p+1 dx
       =nα_n/A-2+n_ℝ^nρ^p+1 dx,

where ρ^* is a rearrangement function of ρ.
Thus we have

    (A-2+n)_ℝ^nρ^p(|x|^A-2*ρ) dx≤ nα_n_ℝ^nρ^p+1 dx +C(A,n,M_0)_ℝ^nρ^p dx.

So, (<ref>) becomes

    d/dt_ℝ^nρ^p dx
    ≤   -4mp(p-1)(m+p-1)^2_ℝ^n|∇ρ^m+p-1/2|^2 dx+(p-1)C(A,n,M_0)_ℝ^nρ^p dx,

which is of the same form as (<ref>) in Theorem <ref>, and the rest of the arguments (including steps 2 and 3 in Theorem <ref>) go through in the same way without any further change.

6pt

Case B.- Next, we deal with the case A=2.

6pt

The formula (<ref>) can be written as

    d/dt_ℝ^nρ^p dx
    =   -4mp(p-1)(m+p-1)^2_ℝ^n|∇ρ^m+p-1/2|^2 dx -nα_n(p-1)_ℝ^nρ^p+1 dx
       +n(p-1)M_0_ℝ^nρ^p dx ,

leading to

    d/dt_ℝ^nρ^p dx
    ≤   -4mp(p-1)(m+p-1)^2_ℝ^n|∇ρ^m+p-1/2|^2 dx +n(p-1)M_0_ℝ^nρ^p dx .

Similar to the first case and the proof of Theorem <ref>, an iterative inequality and the uniform L^∞-bound of solutions can be obtained.




 §.§ The case 2-n< A≤ 2 and λ =0


The main difference is the use of the Hardy-Littlewood-Sobolev (HLS) inequality in order to cope with the different L^p-norm to be bounded. Notice that the case A=2-n was proven in <cit.>, so we reduce to the range 2-n< A≤ 2.


(Space-Time uniform L^∞-bound)
Assume that the initial data ρ_0∈ L^1_+∩ L^∞(ℝ^n), 2-n< A≤ 2, and the diffusion exponent m>1-A/n and λ=0. If ρ is a global weak solution, then it satisfies a uniform in time L^∞-bound

    ρ_L^∞(ℝ_+; L^∞(ℝ^n))≤ C .



 We again proceed in 3 steps.

6pt

Step 1.- We first prove the uniform in time L^p-bound

    ρ^p_L^p≤ρ_0^p_L^p+C(A,n,m,M_0,p)  p>1.

For the case A<2, taking pρ^p-1, p>max{1,A-2+n/2-A}, as a test function in the equation (<ref>), integrating it and using (<ref>), we have

    d/dt_ℝ^nρ^p dx
    =   -4mp(p-1)(m+p-1)^2_ℝ^n|∇ρ^m+p-1/2|^2 dx 
       +(p-1)(A-2+n)_ℝ^nρ^p(|x|^A-2*ρ) dx.

Using the Hardy-Littlewood-Sobolev inequality, see <cit.>, we know that

    _ℝ^n×ℝ^nρ^p(x)ρ(y)/|x-y|^2-A dxdy≤ C_HLSρ^p_L^rρ_L^s,

where 1/r+2-A/n+1/s=2 and the constant C_HLS has an upper bound
 
    C_HLS   ≤n/n-2+A(|S^n-1|/n)^2-A/n1/sr((2-A/n(1-1/s))^2-A/n+(2-A/n(1-1/r))^2-A/n)
       =:C̅(n,A,r,s).

Taking r=p+1/p, then

    s=n(p+1)/(n-2+A)(p+1)+n>0.

Due to p>max{1,A-2+n/2-A}, we have 1<s<p+1. Hence, we deduce

    ρ_L^s≤ρ^1-θ_L^1ρ^θ_L^p+1,  θ=-n+(2-A)(p+1)/np.

Thus (<ref>), (<ref>) and (<ref>) show that

    d/dt_ℝ^nρ^p dx
    ≤   -4mp(p-1)(m+p-1)^2_ℝ^n|∇ρ^m+p-1/2|^2 dx 
       +(p-1)(A-2+n)C_HLSρ^1-θ_L^1ρ^p+θ_L^p+1.

Using that A>2-n and the definition of θ in (<ref>), we can get 1<p+θ<p+1. Notice that

    ρ_L^p+1^p+θ≤ρ_L^1^θ_1(p+θ) (S_n^-1/2∇ρ^m+p-1/2_L^2)^2(p+θ) 1-θ_1/m+p-1,

where θ_1=n(m+p-1)-(p+1)(n-2)/(p+1)(n(m+p-1)-(n-2)).

Due to A<2 and m>1-A/n, we know (1-θ_1)(p+θ)/m+p-1<1. Hence using (<ref>) and the Young inequality, we deduce that

    C(A,n)p^2ρ^1-θ_L^1ρ^p+θ_L^p+1≤    C(σ_1)C(A,n)^q_1p^2q_1 ρ_L^1^(1-θ+θ_1(p+θ))q_1  S_n^-(p+θ)(1-θ_1)q_1/m+p-1
       +σ_1∇ρ^m+p-1/2_L^2^2(p+θ)(1-θ_1)/m+p-1  q_2

for any σ_1>0, C(σ_1)=(σ_1q_1)^-q_1/q_2· q_1^-1, and q_1,q_2>1 satisfy 1/q_1+1/q_2=1.

Given 0<C_1<2mp(p-1)/(m+p-1)^2, we set σ_1=C_1 and 2(p+θ)(1-θ_1)q_2/m+p-1=2, then by (<ref>) and (<ref>), we get

    d/dt∫_ℝ^nρ^pdx≤-C_1∇ρ^m+p-1/2_L^2^2+C(A,m,n,M_0,p).

Hence from (<ref>) and (<ref>), we obtain

    d/dt∫_ℝ^nρ^pdx≤-ρ_L^p^p+C(A,m,n,M_0,p).

A simple computation gives

    ρ_L^p^p≤ρ_0_L^p^p+C(A,m,n,M_0,p)   p>max{1,A-2+n/2-A}.

Together with a simple interpolation inequality, we obtain that the uniform in time L^p-bound in time holds for any p>1.

For the case A=2, then (<ref>) can be written as

    d/dt_ℝ^nρ^p dx
    ≤ -4mp(p-1)(m+p-1)^2_ℝ^n|∇ρ^m+p-1/2|^2 dx +n(p-1)M_0_ℝ^nρ^p dx.

Notice that

    ρ^p_L^p≤ρ^p(1-θ)_L^1ρ^pθ_L^p+1.

Similarly to (<ref>)-(<ref>), we have

    d/dtρ_L^p^p≤ -ρ_L^p^p +C(A,n,m,p,M_0).

Solving the ordinary differential inequality, we obtain the uniform in time L^p-bound for the case A=2.
6pt

Step 2.- In this step, we now prove the following iterative inequality

    d/dtρ_L^p_k^p_k≤-ρ^p_k_L^p_k +C̃ p_k^2ν_2((ρ_L^p_k-1^p_k-1)^η_1+(ρ_L^p_k-1^p_k-1)^η_2),

where 0<η_1,η_2≤ 2, 1<ν_2≤ n+1, p_k=2^k+n/2-A+n (k=0,1,2,...), and C̃ is a constant independent of p_k.

In fact, taking r=p_k+1/p_k and s=n(p_k+1)/(n-2+A)(p_k+1)+n in (<ref>), a similar procedure to prove (<ref>) gives that

    d/dtρ_L^p_k^p_k≤ -2C_1_ℝ^n|∇ρ^m+p_k-1/2|^2 dx +(p_k-1)(A-2+n)C_HLSρ^1-θ_L^1ρ^p_k+θ_L^p_k+1,

where θ =(2-A)(p_k+1)-n/np_k, 0<C_1<2mp_k(p_k-1)/(m+p_k-1)^2 is a constant independent of p_k. Plugging r,s into (<ref>), and after some tedious computations, we deduce

    C_HLS≤C̅(n,A,p_k+1/p_k,n(p_k+1)/(n-2+A)(p_k+1)+n)≤ C(n,A)(p_k+1).

Hence from (<ref>), we have

    d/dtρ_L^p_k^p_k≤ -2C_1_ℝ^n|∇ρ^m+p_k-1/2|^2 dx +(A-2+n)p_k^2ρ^1-θ_L^1ρ^p_k+θ_L^p_k+1.

Using again the interpolation inequality and the Sobolev inequality, we obtain

    ρ_p_k+1≤ρ_L^p_k-1^θ_1ρ^1-θ_1_L^(m+p_k-1)n/n-2≤ρ_L^p_k-1^θ_1(S_n^-1/2∇ρ^m+p_k-1/2_L^2)^2(1-θ_1)/m+p_k-1,

where

    θ_1=1/p_k+1-n-2/n(m+p_k-1)/1/p_k-1-n-2/n(m+p_k-1),
       1-θ_1=1/p_k-1-1/p_k+1/1/p_k-1-n-2/n(m+p_k-1).

Noticing 0<p_k+θ <p_k+1 due to 2-n<A≤ 2, hence we easily compute that

    (1-θ_1)p_k+θ/m+p_k-1<1,   m>1-2/n,  p_k=2^k+n/2-A+n.

Therefore, the Young inequality implies that

    (A-2+n)p_k^2ρ^1-θ_L^1ρ^p_k+θ_L^p_k+1≤     C_1(σ_1)((A-2+n)ρ^1-θ_L^1p_k^2)^ν_2S_n^-ν_2(p_k+θ)(1-θ_1)/m+p_k-1ρ_L^p_k-1^ν_2 (p_k+θ)θ_1,
       +σ_1 ∇ρ^m+p_k-1/2_L^2^2(p_k+θ)(1-θ_1)/m+p_k-1ν_1,

for any σ_1>0, with C_1(σ_1)=(σ_1 ν_1)^-ν_2/ν_1ν_2^-1,
where ν_1, ν_2>1 satisfy 1ν_1+1ν_2=1. Setting

    2(p_k+θ)(1-θ_1)/m+p_k-1ν_1=2         σ_1=C_1 ,

we can compute ν_2=m+p_k-1/(m+p_k-1)-(1-θ_1)(p_k+θ)<n+1.
Hence, (<ref>), (<ref>) and (<ref>) imply

    d/dtρ_L^p_k^p_k≤     -C_1 _ℝ^n |∇ρ^m+p_k-1/2|^2 dx
        + C_1(σ_1)(C(A,B,n,M_0)p_k^2)^ν_2 S_n^-ν_2(p_k+θ)(1-θ_1)/m+p_k-1(ρ^p_k-1_L^p_k-1)^η_2,

where η_2:=ν_2 (p_k+θ)θ_1/p_k-1≤ 2.
Thus, from (<ref>) and (<ref>) we deduce that

    d/dtρ_L^p_k^p_k≤    -ρ_L^p_k^p_k +C(σ_1) S_n^-ℓ_2p_k(1-θ_2)/m+p_k-1(ρ^p_k-1_L^p_k-1)^η_1
       +C_1(σ_1)(C(A,B,n,M_0)p_k^2)^ν_2 S_n^-ν_2(p_k+θ)(1-θ_1)/m+p_k-1(ρ^p_k-1_L^p_k-1)^η_2,

where θ_2, σ_1 and ℓ_2 are same to that in (<ref>) and (<ref>), η_1:=ℓ_2 p_kθ_2/p_k-1≤2.
Noticing that

    C(σ_1)S_n^-ℓ_2p_k(1-θ_2)/m+p_k-1→ C(m,n,A,B,M_0),   p_k→∞,

and

    C_1(σ_1)(C(A,B,n,M_0))^ν_2 S_n^-ν_2(p_k+θ)(1-θ_1)/m+p_k-1→ C(m,n,A,B,M_0),   p_k→∞,

hence there is a constant C̃:=C̃(m,n,A,B,M_0)>1 independent of p_k such that

    max{C(σ_1)S_n^-ℓ_2p_k(1-θ_2)/m+p_k-1,C_1(σ_1)(C(A,B,n,M_0))^ν_2 S_n^-ν_2(p_k+θ)(1-θ_1)/m+p_k-1}≤C̃.

Hence, (<ref>) implies that (<ref>) holds.

6pt

Step 3.- In this step, we will use the iterative inequality (<ref>) to prove the L^∞-bound of ρ by induction.
In (<ref>) at Step 1, we take p=p_0=1+n/2-A+n, then it holds

    ρ^p_0_L^p_0≤ρ_0^p_0_L^p_0+C(A,B,n,m,M_0)≤C̅,

where C̅>1 is a constant independent of p_k and time t.

Let y_k(t):=ρ^p_k_L^p_k, solving the iterative inequality (<ref>), we obtain

    (e^t y_k(t))'≤    C̃ p_k^2ν_2(y_k-1^η_2+y_k-1^η_2)e^t
    ≤    2C̃(2n3-A/2-A)^2(n+1)2^2(n+1)kmax{1,sup_t≥ 0y_k-1^2(t)}e^t,

where the last inequality used the facts p_k=2^k+n/2-A+n≤ 2n3-A/2-A2^k, 0<η_1,η_2≤ 2 and 1<ν_2≤ n+1.
Let a_k:=2C̃(2n3-A/2-A)^2(n+1)2^2(n+1)k>1.
By (<ref>) and (<ref>), we obtain

    y_k(t)≤     a_kmax{1,sup_t≥ 0y_k-1^2(t)}(1-e^-t)+y_k(0)e^-t
    ≤    2 a_kmax{1,sup_t≥ 0y_k-1^2(t),D^2^k}.

From (<ref>) after some iterative steps, we have

    y_k(t)≤    2a_k(2a_k-1)^2(2a_k-2)^2^2···(2a_1)^2^k-1max{sup_t≥ 0y_0^2^k(t),D^2^k}
    
    =    (2C̃(2n3-A/2-A)^2(n+1))^2^k-1 (2^2(n+1))^2·2^k-k-2max{sup_t≥ 0y_0^2^k(t),D^2^k}.

Taking the power 1/p_k to above inequality, then we conclude that

    ρ_L^p_k≤ 2(2n3-A/2-A)^2(n+1) 2^4(n+1)C̃max{sup_t≥ 0y_0(t),D}.

So, (<ref>) and (<ref>) imply that ρ_L^∞ is uniformly bounded in time.







§ L^∞-UNIFORM BOUND FOR THE STRONG SINGULARITY CASE


In this section, we deduce the L^∞-uniform bound of the solutions to the model (<ref>) with -n<B<2-n≤ A≤ 2 for the attractive-repulsive λ>0. The potential |x|^A/A has a well-defined distributional Laplacian as given in the first part of the formula (<ref>):

    Δ(|x|^A/A*ρ)={[ (A-2+n)|x|^A-2*ρ                ,                 ;         nα_nρ(x)                ,                  ].  .

However, the Laplacian of the repulsive part |x|^B/B does not make sense for -n<B<2-n. Hence we utilize a fractional Laplace operator to deal with it.  Let the notation L_s = (-Δ)^s with 0 < s < 1 for the fractional
powers of the Laplace operator define on smooth functions in ℝ^n by Fourier transform
and it is extended in a natural way to functions in the Sobolev space H^2s (ℝ^n).
The inverse operator is denoted by 𝒦_s =(-Δ)^-s and can be realized
by convolution

    𝒦_s u=K_s*u,   K_s=C(n,s)|x|^-(n-2s).

Hence taking s=B+n/2 in (<ref>), we have

    |x|^B/B*u=1/BC(n,B+n/2)(-Δ)^-B+n/2 u.

An important tool in this case is the Stroock-Varopoulos inequality for the fractional Laplace operator L_s = (-Δ)^s, 0 < s < 1, see <cit.> and the references therein,

    _ℝ^n|ρ|^γ-2ρ(-Δ)^α/2ρ dx≥4(γ-1)/γ^2_ℝ^n|(-Δ)^α/4 |ρ|^γ/2|^2 dx,

where 0<α<2, γ>2. We will take the advantage of the repulsive part in this section in order to compensate the attractive nonlocal part by means of the fractional Sobolev inequality <cit.>. As in Section 2, we will take λ=1 in the proof of the next result without loss of generality.


(Space-Time uniform L^∞-bound)
 Assume that the initial data ρ_0∈ L^1_+∩ L^∞(ℝ^n), -n<B<2-n≤ A≤ 2 and the diffusion exponent m>1-2/n. If ρ is a global weak solution, then it satisfies a uniform in time L^∞-bound

    ρ_L^∞(ℝ_+; L^∞(ℝ^n))≤ C ,

for all λ> 0.



Step 1.- We first prove a uniform in time L^p-bound.
Taking pρ^p-1, p>1 as test function in (<ref>), we have

    d/dt_ℝ^nρ^p dx
    =-4mp(p-1)(m+p-1)^2_ℝ^n|∇ρ^m+p-1/2|^2 dx +(p-1)_ℝ^nρ^pΔ (U*ρ) dx.

Now, we focus on the last term in the right-hand side of (<ref>).
For the case 2-n<A< 2, from (<ref>) and (<ref>), we rewrite it as

    _ℝ^nρ^pΔ (U*ρ) dx=    _ℝ^nρ^p(x)(A-2+n)|x|^A-2*ρ(x) dx
       +_ℝ^nρ^p(x)Δ(-C(n,n+B/2)1/B(-Δ)^-B+n/2ρ) dx
    
    =    (A-2+n)_ℝ^nρ^p(x)|x|^A-2*ρ(x) dx
       +C(n,n+B/2)1/B_ℝ^nρ^p(x)(-Δ)^1-B+n/2ρ dx.

Using the Stroock-Varopoulos inequality (<ref>) with γ=p+1, α=2-n-B, then the repulsive term is negative

    1/B_ℝ^nρ^p(x)(-Δ)^1-B+n/2ρ dx≤1/B4p/(p+1)^2_ℝ^n|(-Δ)^2-n-B/4ρ^p+1/2|^2 dx< 0,

since B<0. Therefore, from (<ref>), (<ref>) and (<ref>), we obtain

    d/dt_ℝ^nρ^p dx
    ≤   -4mp(p-1)(m+p-1)^2_ℝ^n|∇ρ^m+p-1/2|^2 dx 
       +(p-1)(A-2+n)_ℝ^n_ℝ^nρ^p(x)ρ(y)/|x-y|^2-A dydx
       +C(n,n+B/2)1/B4p(p-1)/(p+1)^2_ℝ^n|(-Δ)^2-n-B/4ρ^p+1/2|^2 dx.

Using the same method as (<ref>)-(<ref>), we can still obtain that

    (A-2+n)_ℝ^n_ℝ^nρ^p(x)ρ(y)/|x-y|^2-A dydx≤ nα_n_ℝ^nρ^p+1 dx+C(M_0,A,n)_ℝ^nρ^p dx.

Hence (<ref>) can be written as

    d/dt_ℝ^nρ^p dx
    ≤   -4mp(p-1)(m+p-1)^2_ℝ^n|∇ρ^m+p-1/2|^2 dx 
       +(p-1)nα_n_ℝ^nρ^p+1 dx+(p-1)C(M_0,A,n)_ℝ^nρ^p dx
       +C(n,n+B/2)1/B4p(p-1)/(p+1)^2_ℝ^n|(-Δ)^2-n-B/4ρ^p+1/2|^2 dx.

Using the interpolation inequality and the fractional Sobolev inequality <cit.>, we deduce that

    ρ^p+1_L^p+1   ≤ M_0^(1-θ)(p+1)ρ^(p+1)θ_L^n(p+1)/2n-2+B = M_0^(1-θ)(p+1)ρ^p+1/2^2θ_L^2n/2n-2+B
       ≤ S(n,2-n-B/2)^θ M_0^(1-θ)(p+1)(-Δ)^2-n-B/4ρ^p+1/2^2θ_L^2,

where θ=pn/pn+2-n-B, and S(n,s) is given by the following exact form

    S(n,s)=2^-2sπ^-sΓ(n-2s/2)/Γ(n+2s/2)[Γ(n)/Γ(n/2)]^2s/n.


Making use of the Young inequality and the formula (<ref>)-(<ref>), we infer that for any σ_0>0, it holds that

    d/dt_ℝ^nρ^p dx
    ≤   -4mp(p-1)(m+p-1)^2_ℝ^n|∇ρ^m+p-1/2|^2 dx +σ_0_ℝ^n|(-Δ)^2-n-B/4ρ^p+1/2|^2 dx
       +C(n,n+B/2)1/B4p(p-1)/(p+1)^2_ℝ^n|(-Δ)^2-n-B/4ρ^p+1/2|^2 dx
       +(p-1)C(M_0,A,n)_ℝ^nρ^p dx+C(p,n,B,σ_0).

Taking

    σ_0=-C(n,n+B/2)1/B4p(p-1)/(p+1)^2,

we have

    d/dt_ℝ^nρ^p dx
    ≤   -4mp(p-1)(m+p-1)^2_ℝ^n|∇ρ^m+p-1/2|^2 dx
        +(p-1)C(M_0,A,n)_ℝ^nρ^p dx+C(p,n,B,σ_0).

Using the same procedure as (<ref>)-(<ref>), we get

    d/dt_ℝ^nρ^p dx
    ≤ -ρ^p_L^p +C(A,B,n,m,M_0,p).

A simple computation shows that

    ρ^p_L^p≤ρ_0^p_L^p+C(A,B,n,m,M_0,p),

for any fixed p>1, showing that the L^p-norm of ρ is uniformly bounded in time.

For the case A=2, from (<ref>) and (<ref>) it can be obtained that

    d/dt_ℝ^nρ^p dx
    ≤ -4mp(p-1)(m+p-1)^2_ℝ^n|∇ρ^m+p-1/2|^2 dx +n(p-1)M_0_ℝ^nρ^p dx,

which implies the uniform in time L^p-bound for the case A=2.

For the case A=2-n, from (<ref>) and (<ref>) it can be obtained that

    d/dt∫_ℝ^nρ^pdx
    ≤   -4mp(p-1)/(m+p-1)^2∫_ℝ^n|∇ρ^m+p-1/2|^2dx+(p-1)nα(n)∫_ℝ^nρ^p+1(x)dx
       +C(n,n+B/2)1/B4p(p-1)/(p+1)^2_ℝ^n|(-Δ)^2-n-B/4ρ^p+1/2|^2 dx .

Utilizing (<ref>) and the Young inequality, we derive

    d/dt_ℝ^nρ^p dx
    ≤ -4mp(p-1)(m+p-1)^2_ℝ^n|∇ρ^m+p-1/2|^2 dx +C(n,m,p,M_0,A,B),

which implies the uniform in time L^p-bound for the case A=2-n.

6pt

Step 2.- In this step, we now prove the following iterative inequality

    d/dtρ_L^p_k^p_k≤-ρ^p_k_L^p_k +C̃ p_k^ν_2((ρ_L^p_k-1^p_k-1)^η_1+(ρ_L^p_k-1^p_k-1)^η_2),

where 0<η_1,η_2≤ 2, 1<ν_2≤max{n+1,n/2-n-B+1}, p_k=2^k+n (k=0,1,2,...), and C̃ is a constant independent of p_k.

For the case 2-n<A<2, taking p_kρ^p_k-1, p_k=2^k+n>1 as a test function in (<ref>) and using the same procedure to obtain (<ref>), we have

    d/dt_ℝ^nρ^p_k dx
    ≤   -4mp_k(p_k-1)(m+p_k-1)^2_ℝ^n|∇ρ^m+p_k-1/2|^2 dx 
       +(p_k-1)nα(n)_ℝ^nρ^p_k+1 dx+(p_k-1)C(M_0,A,n)_ℝ^nρ^p_k dx
       +C(n,n+B/2)1/B4p_k(p_k-1)/(p_k+1)^2_ℝ^n|(-Δ)^2-n-B/4ρ^p_k+1/2|^2 dx
    ≤   -2C_1_ℝ^n|∇ρ^m+p_k-1/2|^2 dx +C(n,n+B/2)1/2B_ℝ^n|(-Δ)^2-n-B/4ρ^p_k+1/2|^2 dx
       +C(M_0,A,n)(p_k-1)_ℝ^nρ^p_k dx +nα(n)(p_k-1)_ℝ^nρ^p_k+1 dx ,

where 0<C_1<2mp_k(p_k-1)/(m+p_k-1)^2 is a constant independent of p_k.
Using again the interpolation inequality and the fractional Sobolev inequality <cit.>, we obtain

    ρ^p_k+1_p_k+1   ≤ρ_L^p_k-1^(p_k+1)θ_3ρ^(p_k+1)(1-θ_3)_L^n(p_k+1)/2n-2+B
       ≤ S(n,2-n-B/2)^1-θ_3ρ_L^p_k-1^(p_k+1)θ_3(-Δ)^2-n-B/4ρ^p_k+1/2_L^2^2(1-θ_3),

where θ_3=1/p_k+1-2n-2+B/n(p_k+1)/1/p_k-1-2n-2+B/n(p_k+1)∼ O(1). Hence the Young inequality implies that for any σ_3>0
 
    nα_n (p_k-1)ρ^p_k+1_p_k+1≤     C(σ_3)(nα_n S(n,2-n-B/2)^1-θ_3(p_k-1)ρ_L^p_k-1^(p_k+1)θ_3)^q_1
       +σ_3 ((-Δ)^2-n-B/4ρ^p_k+1/2_L^2^2(1-θ_3))^q_2,

where C(σ_3)=(σ_3 q_2)^-q_1/q_2q_1^-1,
where q_1, q_2>1 satisfy 1q_1+1q_2=1. Setting

    (1-θ_3)q_2=1         σ_3=-C(n,n+B/2)1/2B ,

we can compute

    q_1=n(p_k+1)-(2n-2+B)p_k-1/(2-B-n)p_k-1<n/2-n-B+1.

 Therefore using (<ref>), (<ref>) and (<ref>), we deduce

    d/dt_ℝ^nρ^p_k dx
    ≤   -2C_1_ℝ^n|∇ρ^m+p_k-1/2|^2 dx +C(M_0,A,n)p_k_ℝ^nρ^p_k dx
       +C(σ_3)(nα_n S(n,2-n-B/2)^1-θ_3(p_k-1)ρ_L^p_k-1^(p_k+1)θ_3)^q_1,

Using the same process as (<ref>)-(<ref>), we have

    d/dtρ_L^p_k^p_k≤    -ρ_L^p_k^p_k+C(σ_1)(1+(C(A,B,n,M_0)p_k)^ℓ_2) S_n^-ℓ_2p_k(1-θ_2)/m+p_k-1(ρ^p_k-1_L^p_k-1)^ℓ_2 p_kθ_2/p_k-1
       +C(σ_3)(nα_n S(n,2-n-B/2)^1-θ_3(p_k-1)ρ_L^p_k-1^(p_k+1)θ_3)^q_1.

Notice (<ref>) and

    C(σ_3)(nα_n S(n,2-n-B/2)^1-θ_3)^q_1→ C(m,n,A,B,M_0),   p_k→∞,

hence there is a constant C̃:=C̃(m,n,A,B,M_0)>1 independent of p_k such that

    max{C(σ_1)(1+C(A,B,n,M_0)^ℓ_2) S_n^-ℓ_2p_k(1-θ_2)/m+p_k-1,C(σ_3)(nα_n S(n,2-n-B/2)^1-θ_3)^q_1}≤C̃.

Taking 1<ν_2:=max{q_1,l_2}≤max{n+1,n/2-n-B+1} due to (<ref>) and (<ref>), and using the fact that

    η_1:=ℓ_2 p_kθ_2/p_k-1≤ 2 ,  η_2:=p_k+1/p_k-1θ_3q_1=p_k+1/p_k-1≤ 2,

then (<ref>) implies that (<ref>) holds.

It is not difficult to see that the case A=2-n is similar to the case 2-n<A<2. The inequalities above hold true in the limit case A=2-n with some simplications, and an iterative inequality can be also obtained.

Finally, Step 3 in the proof of Theorem <ref> shows the uniform L^∞-bound in time for 2-n≤ A<2.

The case A=2 follows similarly to the first case and the proof of Theorem <ref>, an iterative inequality and the uniform L^∞-bound of solutions can be analogously obtained.








§ ACKNOWLEDGEMENTS

JAC was partially supported by the EPSRC grant number EP/P031587/1.
JW is partially supported  by Program for Liaoning Excellent Talents in University (Grant No. LJQ2015041) and Key Project of Education Department of Liaoning Province (Grant No. LZD201701). The authors warmly thank the referees for the very valuable comments and suggestions that allowed us to improve this paper. We also thank Edoardo Mainini for pointing us out an important reference.

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