Boundedness and large-time behavior for a two-dimensional quasilinear chemotaxis-growth system with indirect signal consumption

Abstract

This paper deals with a two-dimensional quasilinear chemotaxis-growth system with indirect signal consumption

$$\begin{aligned} \left\{ \begin{aligned}&u_t=\nabla \cdot (D(u)\nabla u)-\nabla \cdot (S(u)\nabla v)+\mu (u-u^{2}),&(x,t)\in \Omega \times (0,\infty ), \\&v_t=\Delta v-vw,&(x,t)\in \Omega \times (0,\infty ), \\&w_t=-\delta w+u,&(x,t)\in \Omega \times (0,\infty ), \end{aligned} \right. \end{aligned}$$

under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega \subset {\mathbb {R}}^{2}$, where $\delta >0$ and $\mu >0$, the nonlinear diffusivity D(u) and chemosensitivity S(u) are supposed to satisfy

$$\begin{aligned} D(u)\geqslant u^{m},\quad S(u)\leqslant u^{q}\quad \text {and}\quad D,S>0. \end{aligned}$$

When $m>\max \{2q-3, 1\}$, we study the global boundedness of solutions for this problem. In the case of $m=0$ and $0<q<\frac{3}{2}$, we can obtain the boundedness of this system by using different methods. Moreover, under the particular conditions of $m=0$ and $q=1$, the global bounded solution (u, v, w) satisfies

$$\begin{aligned} \Vert u(\cdot ,t)-1\Vert _{L^{\infty }(\Omega )}+\Vert v(\cdot ,t)\Vert _{L^{\infty }(\Omega )}+\Vert w(\cdot ,t)-\frac{1}{\delta }\Vert _{L^{\infty }(\Omega )}\rightarrow 0 \end{aligned}$$

as $t\rightarrow \infty$.