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

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 χ > 0, µ > 0, $r \geqslant 2$ 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 regularising influence of its associated diffusion operator. In both constructions, we naturally rely on the regularising properties of a sufficiently strong logistic source term in the first equation.