$\u211d^N$上带多孔介质扩散、线性产生项与Logistic源的趋化模型弱解的全局存在性
Global existence of weak solutions to chemotaxis models with porous medium diffusion, linear production, and logistic source on $\mathbb{R}^N$
中文总结 AI 辅助
该文研究$\u211d^N$上带多孔介质扩散、线性产生项与Logistic源的趋化系统，给出不同参数范围下弱解的全局存在性、有界性条件，并证明特定范围内Hölder连续弱解的唯一性。
中文摘要 AI 辅助
本文研究如下趋化系统弱解的全局可解性、有界性与唯一性：\n$$\begin{cases} u_t = \Delta u^m - \chi\nabla \cdot (u \nabla v) + u(a - b u), & \text{在 } (0,\infty)\times\mathbb{R}^N\text{ 中}, \\\\ \tau v_t = \Delta v - \lambda v + \mu u, & \text{在 } (0,\infty)\times\mathbb{R}^N\text{ 中}, \end{cases}$$\n其中$m>1$，$\tau\in\{0,1\}$，$\lambda,\mu,a,b>0$，$\chi\in\mathbb{R}$。对任意$m>1$，我们证明了初始数据不必可积时弱解的存在性，尽管这类解通常未必有界。随后我们表明，在抛物-抛物情形（$\tau=1$）下，当$m>\frac{2N}{N+2}$时存在全局有界弱解；当$1<m\le \frac{2N}{N+2}$时，若Logistic阻尼系数$b$足够大，也存在全局有界弱解。在抛物-椭圆情形（$\tau=0$）下，我们证明当$m>2-\frac{2}{N}$时存在全局有界弱解；当$1<m\le 2-\frac{2}{N}$时，若$b$足够大，同样存在全局有界弱解。最后，对于$1<m\le 3$，我们证明了在初始时刻仍保持Hölder连续的弱解的唯一性。
英文摘要
This paper investigates the global solvability, boundedness, and uniqueness of weak solutions to the chemotaxis system \begin{equation*} \begin{cases} u_t = Δu^m - χ\nabla \cdot (u \nabla v) + u(a - b u), & \text{in } (0,\infty)\times\mathbb{R}^N, \\ τv_t = Δv - λv + μu, & \text{in } (0,\infty)\times\mathbb{R}^N, \end{cases} \end{equation*} where \(m>1\), \(τ\in\{0,1\}\), \(λ,μ,a,b>0\), and \(χ\in\mathbb{R}\). For every \(m>1\), we establish the existence of weak solutions for initial data that are not necessarily integrable, although such solutions need not be bounded in general. We then show that globally bounded weak solutions exist in the parabolic-parabolic case \((τ=1)\) when \(m>\frac{2N}{N+2}\), and also when \(1<m\le \frac{2N}{N+2}\) provided that the logistic damping coefficient \(b\) is sufficiently large. In the parabolic-elliptic case \((τ=0)\), we prove the existence of globally bounded weak solutions when \(m>2-\frac{2}{N}\), and also when \(1<m\le 2-\frac{2}{N}\) provided that \(b\) is sufficiently large. Finally, for \(1<m\le 3\), we prove uniqueness of weak solutions that are Hölder continuous up to the initial time.
