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

TL;DR

This paper establishes the existence of weak solutions for a degenerate haptotaxis system with a potentially degenerate diffusion tensor in two and three dimensions, using logistic growth to aid regularization.

Contribution

It introduces new methods to construct weak solutions under mild conditions on the diffusion tensor, including degenerate cases, and also constructs classical solutions when the diffusion tensor is strictly positive.

Findings

Existence of weak solutions under mild assumptions on the diffusion tensor.

Construction of classical solutions with positive definite diffusion tensor.

Utilization of logistic source term for regularization and solution construction.

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}) - \chi \nabla \cdot (u\mathbb{D}\nabla w) + \mu u(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 $\chi > 0$, $\mu > 0$, $r \geq 2$ and $\mathbb{D}: \overline{\Omega} \rightarrow \mathbb{R}^{n\times n}$, $\mathbb{D}$ 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.