Boundedness of solutions to a quasilinear higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion

Jiashan Zheng

doi:10.3934/dcds.2017026

Article Contents

2017, Volume 37, Issue 1: 627-643. Doi: 10.3934/dcds.2017026

This issue Previous Article Bound state solutions of Schrödinger-Poisson system with critical exponent Next Article Blow-up of solutions to the periodic generalized modified Camassa-Holm equation with varying linear dispersion

Boundedness of solutions to a quasilinear higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion

Jiashan Zheng^,

School of Mathematics and Statistics Science, Ludong University, Yantai 264025, China

* Corresponding author: zhengjiashan2008@163.com
* Corresponding author: zhengjiashan2008@163.com

Received: January 2016

Revised: September 2016

Published: September 2017

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Abstract

This paper deals with the Neumann problem for the coupled quasilinear chemotaxis-haptotaxis model of cancer invasion given by

$\left\{ \begin{gathered} ut = \nabla \cdot \left( {{{\left( {u + 1} \right)}^{m - 1}}\nabla u} \right) - \nabla \cdot \left( {u\nabla v} \right) - \nabla \cdot \left( {u\nabla w} \right) + u\left( {1 - u - w} \right), \hfill \\ ut = \Delta v - v + u, \hfill \\ wt = - vw, \hfill \\ \end{gathered} \right.$

where the parameter $m≥q1$ and $\mathbb{R}^N(N≥q2)$ is a bounded domain with smooth boundary. If $m>\frac{2N}{N+2}$, then for any sufficiently smooth initial data there exists a classical solution which is global in time and bounded. The results of this paper partly extend previous results of several authors.

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Mathematics Subject Classification: Primary:92C17, 35K55;Secondary:35K59, 35K20.

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