具有潜在退化扩散张量的二维和三维趋触系统的整体解
Global solutions to a haptotaxis system with a potentially degenerate diffusion tensor in two and three dimensions
摘要：
考虑光滑有界区域$Ω\subseteq \mathbb{R}^n$，$n \in \{2,3\}$中的潜在退化趋触系统\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*}，\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*}，$w_0$，$u_0$，$χ> 0$，$μ> 0$，$r \geq 2$和$\mathbb{D}: \overlineΩ \rightarrow \mathbb{R}^{n\times n}$，$\mathbb{D}$在$\overlineΩ$上半正定.关于上述系统，我们的主要结果是在相当温和的假设下，$\mathbb{D}$以及初始数据的弱解的建设，包括方案的退化扩散的第一个方程。作为在这个建设的一个步骤，以及潜在的独立利益的结果，我们进一步构造经典的解决方案，为同一系统的$\mathbb{D}$的全局正性假设下，确保其相关的扩散算子的充分正则化的影响。在这两种构造中，我们自然地依赖于第一个方程中足够强的logistic源项的正则化性质。
英文摘要：
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.
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