论文标题
全球存在，唯一性和$ l^{\ infty} $ - 分数时间空间keller-segel系统的弱解决方案绑定
Global existence, uniqueness and $L^{\infty}$-bound of weak solutions of fractional time-space Keller-Segel system
论文作者
论文摘要
本文研究了一类时空分数抛物线 - 椭圆形凯勒 - segel方程的弱解决方案的特性，具有逻辑源项，$ \ mathbb {r}^{n} $，$ n \ geq 2 $。建立了全局存在和$ l^{\ infty} $ - 弱解决方案的绑定。我们主要将阻尼系数分为两种情况：（i）$ b> 1- \fracα{n} $，以任何初始值和出生率； （ii）$ 0 <b \ leq 1- \fracα{n} $，用于较小的初始价值和小出生率。通过验证对构造的正则化方程的解决方案的存在，并结合了时间分数部分偏微分方程的广义紧凑性标准来获得存在结果。同时，我们通过建立分数差分不平等并使用Moser迭代方法来获得$ l^{\ infty} $限制的弱解决方案。此外，我们通过在阻尼系数很强时使用超缩估计来证明弱溶液的独特性。最后，我们还为弱解决方案提出了一个爆炸标准，也就是说，如果在有限的时间内使用弱解决方案，那么对于所有$ h> q $，$ l^{h} $ - 弱解决方案的规范同时爆炸。
This paper studies the properties of weak solutions to a class of space-time fractional parabolic-elliptic Keller-Segel equations with logistic source terms in $\mathbb{R}^{n}$, $n\geq 2$. The global existence and $L^{\infty}$-bound of weak solutions are established. We mainly divide the damping coefficient into two cases: (i) $b>1-\fracα{n}$, for any initial value and birth rate; (ii) $0<b\leq 1-\fracα{n}$, for small initial value and small birth rate. The existence result is obtained by verifying the existence of a solution to the constructed regularization equation and incorporate the generalized compactness criterion of time fractional partial differential equation. At the same time, we get the $L^{\infty}$-bound of weak solutions by establishing the fractional differential inequality and using the Moser iterative method. Furthermore, we prove the uniqueness of weak solutions by using the hyper-contractive estimates when the damping coefficient is strong. Finally, we also propose a blow-up criterion for weak solutions, that is, if a weak solution blows up in finite time, then for all $h>q$, the $L^{h}$-norms of the weak solution blow up at the same time.
