分数阶趋化模型在临界Besov空间中解的整体存在性

Global Existence of Solutions for the Fractional Chemotaxis Model in Critical Besov Spaces

该文主要研究如下的分数阶趋化模型:{■_(t)+(-△)^(α/2)=▽·(u▽v)(x,t)∈R^(n)×(0,∞),ε■_(t)v+(-△)^(β/2)v=u,(x,t)∈R^(n)×(0,∞),u(x,0)=u_(0)(x),v(x,0)=v_(0)(x),x∈R^(n)其中α∈[1,2],β∈(0,2],ε≥0.基于分数阶耗散方程在Chemin-Lerner混合时空空间中的线性估计和Fourier局部化方法,作者得到了如下结果:(1)当ε=0时,建立了次临界情形1<α≤2下该模型在Besov空间中的局部适定性和小初值问题的整体适定性,优化了[陈化,吕文斌,吴少华.分数阶趋化模型在Besov空间中解的存在性.中国科学:数学,2019,49(12):1-17]所得适定性结果中正则性和可积性指标的范围.并且还建立了临界情形α=1下该模型在Besov空间中小初值问题的整体适定性;(2)当ε>0时,利用特殊的迭代技巧,作者分别建立了次临界情形1<α≤2和临界情形α=1下该模型在Besov空间中的局部适定性和小初值问题的整体适定性.进一步,利用模型所特有的代数结构,作者还证明了对初值v0无小性条件下解的整体存在性.

In this paper,the authors are concerned with the Cauchy problem of the following fractional chemotaxis model:{■_(t)+(-△)^(α/2)=▽·(u▽v)(x,t)∈R^(n)×(0,∞),ε■_(t)v+(-△)^(β/2)v=u,(x,t)∈R^(n)×(0,∞),u(x,0)=u_(0)(x),v(x,0)=v_(0)(x),x∈R^(n)whereα∈[1,2],β∈(0,2],ε≥0.Based on the linear estimates of the fractional dissipative equation in Chemin-Lerner mixed time-space spaces and the Fourier localization argument,they give the following results:(1)In the case ofε=0,the authors establish the local wellposedness and global well-posedness with small initial data for the subcritical chemotaxis model(1<α≤2),which improves the regularity and integrability indices of the wellposedness results in[Chen,H.,Lv,W.B.,and Wu.S.H.,Existence for a class of chemotaxis model with fractional diffusion in Besov spaces,Sci.Sin.Math.,2019,49(12):1-17(in Chinese)]to more extensive range.Moreover,the authors also establish the global existence of solutions with small initial data for the critical chemotaxis model(α=1);(2)In the case ofε>0,by using a special iteration argument,the authors establish the global wellposedness of this chemotaxis model with small initial data in Besov spaces for the subcritical case(α∈(1,2])and the critical case(α=1),respectively.Furthermore,by using certain algebraical structure of equations,the authors prove the global existence of solutions without smallness assumption imposed on initial data v0.