Global existence and uniform boundedness of smooth solutions to a parabolic-parabolic chemotaxis system with nonlinear diffusion

Abstract

This paper is devoted to the following quasilinear chemotaxis system: \(\bigl\{\scriptsize{ \begin{array}{l} u_{t}=\nabla\cdot(D(u)\nabla u)-\nabla\cdot(u\chi(v)\nabla v)+uf(u),\quad x\in\Omega,t>0, \\ v_{t}=\Delta v-ug(v),\quad x\in\Omega,t>0, \end{array} }\bigr. \) under homogeneous Neumann boundary conditions in a smooth bounded domain \(\Omega\subset\mathbb{R}^{N}\). The given functions \(D(s)\), \(\chi(s)\), \(g(s)\), and \(f(s)\) are assumed to be sufficiently smooth for all \(s\ge0\) and such that \(f(s)\le\kappa-\mu s^{\tau}\). It is proved that the corresponding initial boundary value problem possesses a unique global classical solution for any \(\mu>0\) and \(\tau \ge1\), which is uniformly bounded in \(\Omega\times(0,+\infty)\). Moreover, when \(\kappa=0\), the decay property of the solution is also discussed in this paper.