Global Solvability in a Two-Species Chemotaxis System with Signal Production

Abstract

In this paper, we study the following prototypical two-species chemotaxis system with Lotka-Volterra competition and signal production: $$ \left \{ \textstyle\begin{array}{l@{\quad }l} u_{t}=\Delta u-\nabla \cdot (u\nabla w)+u(1-u^{\theta -1}-v), &x\in \Omega , t>0, \\ v_{t}=\Delta v-\nabla \cdot (v\nabla w)+v(1-v-u), & x\in \Omega , t>0, \\ w_{t}=\Delta w-w+ u+ v, & x\in \Omega , t>0. \end{array}\displaystyle \right .(\ast ) $$ We show that if $\theta >1+\frac{N-2}{N}$ , the associated Neumann initial-boundary value problem for (∗) admits a global generalized solutions in a bounded and smooth domain $\Omega \subset \mathbb{R}^{N}$ $(N\geq 2)$ under merely integrable initial data.