赵继红,李秀蓉.分数阶趋化模型在临界Besov空间中解的整体存在性*[J].数学年刊A辑,2022,43(4):367~386
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
投稿时间:2021-09-24  修订日期:2022-10-02
DOI:10.16205/j.cnki.cama.2022.0024
中文关键词:  分数阶趋化模型, 整体解, Besov空间
英文关键词:Fractional chemotaxis model, Global solutions, Besov spaces
基金项目:国家自然科学基金 (No.11961030), 陕西省自然科学基础研究计划-面上项目(No.2022JM-034) 和宝鸡文理学院人才引进项目(No.209040020)
作者单位
赵继红 宝鸡文理学院数学与信息科学学院, 陕西 宝鸡 721013. 
李秀蓉 西北农林科技大学理学院, 陕西 杨凌 712100. 
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中文摘要:
      该文主要研究如下的分数阶趋化模型:?tu + (??) α/2 u = ? · (u?v), (x, t) ∈ Rn × (0, ∞), ε?tv + (??) β/2 v = u, (x, t) ∈ Rn × (0, ∞), u(x, 0) = u0(x), v(x, 0) = v0(x), x ∈ Rn,基于α ∈ [1, 2], β ∈ (0, 2], ε > 0.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 t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.
英文摘要:
      In this paper, the authors are concerned with the Cauchy problem of the following fractional chemotaxis model:?tu + (??) α/2 u = ? · (u?v), (x, t) ∈ Rn × (0, ∞), ε?tv + (??) β/2 v = u, (x, t) ∈ Rn × (0, ∞), u(x, 0) = u0(x), v(x, 0) = v0(x), x ∈ Rn,where α ∈ [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 of ε = 0, the authors establish the local wellposedness and global well-posedness with small initial data for the subcritical chemotaxis model (1 < α 6 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 of ε > 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.
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