Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Annales Polonici Mathematici / All issues

Abstract

The purpose of this paper is to study the chemotaxis growth system $$\begin{cases} u_t=\Delta u-\nabla \cdot (u|\nabla v|^{p-2}\nabla v)+au-bu^{\alpha},& x\in \Omega ,\, t \gt 0,\\ v_t=\Delta v-v+w,& x\in \Omega ,\, t \gt 0,\\ w_t=\Delta w-w+u,& x\in \Omega ,\, t \gt 0, \end{cases}$$ in a smooth bounded domain $\Omega \subset \mathbb R^n$, $n\geq 2$ with nonnegative initial data and homogeneous boundary conditions of Neumann type for $u,v$ and $w$. We will show that the problem admits a global weak solution when $p\in (1,\frac{n\alpha +2n-6\alpha +4}{2n-6\alpha +4})$, $3\alpha -2\leq n\leq 4\alpha -2$, and when $p \gt 1$, $n \lt 3\alpha -2$. What is more, under appropriate conditions, this global solution with nonnegative initial data $(u_0,v_0,w_0)$ eventually becomes a classical solution of the system and satisfies $$\begin{align*} u\rightarrow (a_{+}/b)^{\frac{1}{\alpha -1}},\quad v\rightarrow (a_{+}/b)^{\frac{1}{\alpha -1}},\quad w\rightarrow(a_{+}/b)^{\frac{1}{\alpha -1}}\quad\ \text{in}\quad L^{\infty}(\Omega), \end{align*}$$ as $t\rightarrow \infty.$