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arXiv 2008.01887math.APmath.DS
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任意维情形下具有奇异敏感度的抛物-椭圆趋化模型中 logistic 源对有限时间爆破的阻止

Finite-time blow-up prevention by logistic source in parabolic-elliptic chemotaxis models with singular sensitivity in any dimensional setting

Halil Ibrahim Kurt, Wenxian Shen

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AI总结：

本文研究任意维奇异敏感度抛物-椭圆趋化系统，证明强 logistic 源可保证经典解全局存在、阻止有限时间爆破，并给出一致有界及 σ=0 情形条件。

摘要：

近年来，logistic 动力学是否足以保证各种趋化模型中经典解的整体存在或阻止有限时间爆破，引起了广泛关注。本文旨在任意空间维情形下，针对下述具有奇异敏感度和 logistic 源的抛物-椭圆趋化系统研究上述问题：给出耦合方程组，其中 u_t=Δu−χ∇·((u/v)∇v)+u(a(x,t)−b(x,t)u^{1+σ})，0=Δv−μv+νu，x∈Ω，并在光滑边界 ∂Ω 上满足 ∂u/∂n=∂v/∂n=0；Ω⊂R^n 为具有光滑边界的有界区域，χ 为奇异趋化敏感度系数，a(x,t) 与 b(x,t) 为正光滑函数，μ、ν 为正常数，σ≥0。当 σ>0 时，我们证明，对任意给定的非负非零初值 u_0∈C^0(\bar Ω)，系统 (0.1) 都存在唯一的全局经典解 (u_σ(x,t;u_0),v_σ(x,t;u_0))，且 u_σ(x,0;u_0)=u_0(x)。这表明在任意空间维情形下，即使 χ 任意大，强 logistic 动力学也足以保证经典解整体存在，从而阻止有限时间爆破发生。此外，在条件 a_inf 大于 μχ²/4（当 0<χ≤2）或 μ(χ−1)（当 χ>2）时，证明解一致有界。当 σ=0 时，我们表明，只要 a(x,t) 和 u_0(x) 都不小，经典解 (u(x,t;u_0,0),v(x,t;u_0,0)) 即全局存在并保持有界。

英文摘要：

In recent years, a lot of attention has been drawn to the question of whether logistic kinetics is sufficient to enforce the global existence of classical solutions or to prevent finite-time blow-up in various chemotaxis models. The current paper is to study the above question for the following parabolic-elliptic chemotaxis system with singular sensitivity and logistic source in any space dimensional setting, \begin{equation} \begin{cases} u_t=Δu-χ\nabla\cdot (\frac{u}{v} \nabla v)+u(a(x,t)-b(x,t) u^{1+σ}),\quad &x\in Ω\cr 0=Δv-μv+νu,\quad &x\in Ω\quad \cr\frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=0,\quad &x\in\partialΩ, \end{cases} \end{equation} where $Ω\subset \mathbb{R}^n$ is a bounded domain with smooth boundary $\partialΩ$, $χ$ is the singular chemotaxis sensitivity coefficient, $a(x,t)$ and $b(x,t)$ are positive smooth functions, $μ,ν$ are positive constants, and $σ\ge 0$. When $σ>0$, we prove that, for every given nonnegative initial data $0\not\equiv u_0\in C^0(\bar Ω)$, (0.1) has a unique globally defined classical solution $(u_σ(x,t;u_0),v_σ(x,t;u_0))$ with $u_σ(x,0;u_0)=u_0(x)$, which shows that, in any space dimensional setting, strong logistic kinetics is sufficient to enforce the global existence of classical solutions and hence prevents the occurrence of finite-time blow-up even for arbitrarily large $χ$. In addition, the solutions are shown to be uniformly bounded under the conditions \begin{equation*} a_{\inf}> \begin{cases} \frac{μχ^2}{4}, &\text{if $0< χ\leq 2,$}\\ μ(χ-1), &\text{if $χ>2$.}\\ \end{cases} \end{equation*} When $σ=0$, we show that the classical solution $(u(x,t;u_0,0),v(x,t;u_0,0))$ exists globally and stays bounded provided that both $a(x,t)$ and $u_0(x)$ are not small.

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