$L^{\infty }$ bounds and asymptotic behavior in a doubly degenerate chemotaxis system below six dimensions

Abstract

We investigate a doubly degenerate nutrient-taxis system of the form \begin{equation*} \begin{cases} u_t = \nabla \cdot (u v \nabla u) - χ\nabla \cdot (u^αv \nabla v) + \ell u v, \qquad &x \in Ω, \ t > 0, v_t = Δv - u v, \qquad &x \in Ω, \ t > 0, \end{cases} \end{equation*} subject to the homogeneous Neumann boundary conditions in a smoothly bounded convex domain $\Omega \subset {\mathbb{R}}^n$ with $n\in \left\{3,4,5\right\}$, where $\alpha \ge 1$, $\chi >0$ and $\ell \ge 0$. For any suitably regular initial data, we establish the global existence of a weak solution that remains uniformly bounded in time, provided that $\alpha$ lies in the range $\left[1,\frac{5}{2}-\frac{n}{4}\right)$, and we also determine the large-time behavior of these solutions. Our proof relies on several novel functional inequalities, a bootstrap argument, and a Moser iteration method.