Global boundedness and decay for a multi-dimensional chemotaxis-haptotaxis system with nonlinear diffusion

Pan  Zheng

doi:10.3934/dcdsb.2016035

Article Contents

2016, Volume 21, Issue 6: 2039-2056. Doi: 10.3934/dcdsb.2016035

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Global boundedness and decay for a multi-dimensional chemotaxis-haptotaxis system with nonlinear diffusion

Pan Zheng^1,

1.
Department of Applied Mathematics Chongqing University of Posts, and Telecommunications, Chongqing 400065

Received: July 31, 2015

Revised: October 31, 2015

Published: July 2016

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Abstract

This paper deals with a parabolic-elliptic-ODE chemotaxis-haptotaxis system with nonlinear diffusion \begin{eqnarray*}\label{1a} \left\{ \begin{split}{} &u_t=\nabla\cdot(\varphi(u)\nabla u)-\chi\nabla\cdot(u\nabla v)-\xi\nabla\cdot(u\nabla w)+\mu u(1-u-w), \\ &0=\Delta v-v+u, \\ &w_{t}=-vw, \end{split} \right. \end{eqnarray*} under Neumann boundary conditions in a smooth bounded domain $\Omega\subset \mathbb{R}^{n}$ $(n\geq1)$, where $\chi$, $\xi$ and $\mu$ are positive parameters and $\varphi(u)$ is a nonlinear diffusion. Under the non-degenerate diffusion and some suitable assumptions on positive parameters $\chi,\xi,\mu$, it is shown that the corresponding initial boundary value problem possesses a unique global classical solution that is uniformly bounded in $\Omega\times(0,\infty)$. Moreover, under the degenerate diffusion, it is proved that the corresponding problem admits at least one nonnegative global bounded-in-time weak solution. Finally, for the suitably small initial data $w_{0}$, we give the decay estimate of $w$.

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Mathematics Subject Classification: Primary: 35K55; Secondary: 35B45, 35B33, 35K57, 92C17.

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