Global solutions to a haptotaxis system with a potentially degenerate diffusion tensor in two and three dimensions

Abstract

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 $\Omega \subseteq {\mathbb{R}}^n$, $n\in \{2,3\}$, with a no-flux boundary condition, positive initial data $u_0$, $w_0$ and parameters $\chi >0$, $\mu >0$, $r\ge 2$ and $\mathbb{D}:\overline{\Omega }\to {\mathbb{R}}^{n\times n}$, $\mathbb{D}$ positive semidefinite on $\overline{\Omega }$. 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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