六维以下双重退化趋化系统的$L^\textit{∞}$界与渐近行为
$L^\infty$ bounds and asymptotic behavior in a doubly degenerate chemotaxis system below six dimensions
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中文总结 AI 辅助
针对三维至五维的双重退化营养趋化系统，该研究证明了特定参数范围内全局一致有界弱解的存在性及长时间行为，所用方法含新型函数不等式、bootstrap论证与Moser迭代法。
中文摘要 AI 辅助
我们研究一类双重退化营养趋化系统，其形式为：$\begin{cases} u_t = \nabla \cdot (u v \nabla u) - \chi \nabla \cdot (u^\alpha v \nabla v) + \ell u v, \quad &x \in \Omega, \\ t > 0, \\\\ v_t = \Delta v - u v, \quad &x \in \Omega, \\ t > 0, \end{cases}$，该系统在光滑有界凸域$\Omega \subset \mathbb{R}^n$中满足齐次Neumann边界条件，其中$n \in \{3,4,5\}$，$\alpha \geq 1$，$\chi>0$，$\ell \geq 0$。对于任意适当正则的初始数据，当$\alpha$处于$\left[1, \frac{5}{2} - \frac{n}{4}\right)$范围内时，我们证明了该系统存在全局弱解，且该弱解随时间保持一致有界；同时确定了这些解的长时间行为。我们的证明依赖于若干新型函数不等式、 bootstrap 论证以及 Moser 迭代法。
英文摘要
We investigate a doubly degenerate nutrient-taxis system of the form \begin{equation*} \begin{cases} u_t = \nabla \cdot (u v \nabla u) - χ\nabla \cdot (u^αv \nabla v) + \ell u v, \qquad &x \in Ω, \ t > 0, v_t = Δv - u v, \qquad &x \in Ω, \ t > 0, \end{cases} \end{equation*} subject to the homogeneous Neumann boundary conditions in a smoothly bounded convex domain $Ω\subset \mathbb{R}^n$ with $n\in \left \{ 3,4,5 \right \}$, where $α\geq 1$, $χ>0$ and $\ell \geq 0$. For any suitably regular initial data, we establish the global existence of a weak solution that remains uniformly bounded in time, provided that $α$ lies in the range $\left[1, \frac{5}{2} - \frac{n}{4}\right)$, and we also determine the large-time behavior of these solutions. Our proof relies on several novel functional inequalities, a bootstrap argument, and a Moser iteration method.
