论文标题
在两个维度和三个维度上具有退化扩散张量的触觉系统的全球解决方案
Global solutions to a haptotaxis system with a potentially degenerate diffusion tensor in two and three dimensions
论文作者
论文摘要
我们认为潜在的退化haptotaxis系统\ begin {equation*} \ left \ { \ begin {Aligned} u_t＆= \ nabla \ cdot（\ Mathbb {d} \ nabla u + u \ u \ nabla \ cdot \ cdot \ cdot \ mathbb {d}） - χ\ nabla \ cdot（u \ mathbb {d} w_t＆= - UW \ end {Aligned} \ right。 \ end {equation*}在平稳的界面$ω\子群中$ \ mathbb {d}：\OverlineΩ\ rightarrow \ Mathbb {r}^{n \ times n} $，$ \ mathbb {d} $ pastic $ pastic semide semidefinite on $ \overlineΩ$。 关于上述系统，我们的主要结果是在相当温和的假设上构建弱解决方案，并在$ \ mathbb {d} $上以及初始数据，其中包括第一个方程式中退化扩散的情况。作为该构建的一步以及潜在的独立兴趣的结果，我们在$ \ Mathbb {D} $的全球阳性假设下进一步构建了同一系统的经典解决方案，从而确保其相关扩散操作员的全面正规化影响。在这两种结构中，我们自然都依赖于第一个方程中足够强的逻辑源项的正规化属性。
We consider the potentially degenerate haptotaxis system \begin{equation*} \left\{ \begin{aligned} u_t &= \nabla \cdot (\mathbb{D} \nabla u + u \nabla \cdot \mathbb{D}) - χ\nabla \cdot (u\mathbb{D}\nabla w) + μu(1-u^{r- 1}), \\ w_t &= - uw \end{aligned} \right. \end{equation*} in a smooth bounded domain $Ω\subseteq \mathbb{R}^n$, $n \in \{2,3\}$, with a no-flux boundary condition, positive initial data $u_0$, $w_0$ and parameters $χ> 0$, $μ> 0$, $r \geq 2$ and $\mathbb{D}: \overlineΩ \rightarrow \mathbb{R}^{n\times n}$, $\mathbb{D}$ positive semidefinite on $\overlineΩ$. Our main result regarding the above system is the construction of weak solutions under fairly mild assumptions on $\mathbb{D}$ as well as the initial data, encompassing scenarios of degenerate diffusion in the first equation. As a step in this construction as well as a result of potential independent interest, we further construct classical solutions for the same system under a global positivity assumption for $\mathbb{D}$, which ensures the full regularizing influence of its associated diffusion operator. In both constructions, we naturally rely on the regularizing properties of a sufficiently strong logistic source term in the first equation.
