Global existence and decay for a chemotaxis-growth system with generalized volume-filling effect and sublinear secretion

Abstract

This paper deals with a fully parabolic chemotaxis-growth system with generalized volume-filling effect and sublinear secretion $$\begin{aligned} \left\{ \begin{array}{ll} u_t=\nabla \cdot (\varphi (u)\nabla u)-\nabla \cdot (\psi (u)\nabla v)+ru-\mu u^{2}, &{}\quad (x,t)\in \Omega \times (0,\infty ), \\ v_{t}=\Delta v-v+g(u), &{}\quad (x,t)\in \Omega \times (0,\infty ), \end{array} \right. \end{aligned}$$ under homogeneous Neumann boundary conditions in a smooth bounded domain \(\Omega \subset \mathbb {R}^{2}\), where \(\varphi (u)\) is a nonlinear diffusion function, \(\psi (u)\) is a chemotactic sensitivity and g(u) is a production rate of the chemoattractant. Under some suitable assumptions on the nonlinearities \(\varphi (u)\), \(\psi (u)\) and g(u), we study the global boundedness and decay of solutions for the system.