Large-Data Global Generalized Solutions in a Chemotaxis System with Tensor-Valued Sensitivities

Abstract

The chemotaxis system $u_t=\Delta u - \nabla \cdot (uS(x,u,v)\cdot\nabla v); v_t=\Delta v - uf(v)$ (referred to as ($\star$) in this abstract), for the density $u=u(x,t)$ of a cell population and the concentration $v=v(x,t)$ of an attractive chemical consumed by the former, is considered under no-flux boundary conditions in a bounded domain $\Omega\subset{\mathbb{R}}^n$, $n\ge 1$, with smooth boundary, where $f \in C^1([0,\infty);[0,\infty))$ and $S \in C^2(\bar\Omega\times [0,\infty)^2;{\mathbb{R}}^{n\times n})$ are given functions such that f(0)=0. In contrast to related Keller--Segel-type problems with scalar sensitivities, in the presence of such matrix-valued $S$ the system ($\star$) in general apparently does not possess any useful gradient-like structure. Accordingly, its analysis needs to be based on new types of a priori bounds. Using a spatio-temporal $L^2$ estimate for $\nabla \ln (u+1)$ as a starting point, we derive a series of compactness properties of solutions to suitably regularized vers...