Boundedness in a two-dimensional doubly degenerate nutrient taxis system

Abstract

In this work, we study the no-flux initial-boundary value problem for the doubly degenerate nutrient taxis system \begin{align} \begin{cases}\tag{\(\star\)}\label{eq 0.1} u_t=\nabla \cdot(u v \nabla u)-\chi \nabla \cdot\left(u^{2} v \nabla v\right)+\ell u v, & x \in \Omega, t>0, \\ v_t=\Delta v-u v, & x \in \Omega, t>0 \end{cases} \end{align} in a smoothly bounded convex domain \(\Omega \subset \mathbb{R}^2\), where \(\chi>0\) and \(\ell \geq 0\). In this paper, we present that for all reasonably regular initial data, the model \eqref{eq 0.1} possesses a global bounded weak solution which is continuous in its first and essentially smooth in its second component. \end{abstract}