Boundedness in a flux-limited chemotaxis-haptotaxis model with nonlinear diffusion

Hui Wang; Pan Zheng; Runlin Hu

doi:10.3934/eect.2023004

Article Contents

2023, Volume 12, Issue 4: 1133-1144. Doi: 10.3934/eect.2023004 open access

This issue Previous Article A new class of small initial data which may shift the critical power and lifespan estimates for the classical damped wave equations Next Article A coupled problem described by time-dependent subdifferential operator and non-convex perturbed sweeping process

Boundedness in a flux-limited chemotaxis-haptotaxis model with nonlinear diffusion

1.
College of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
2.
Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, China
3.
School of Mathematics and Statistics, Yunnan University, Kunming 650091, China

^* Corresponding author: Pan Zheng
^* Corresponding author: Pan Zheng

Received: September 2022

Early access: February 2023

Published: August 2023

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Abstract

This paper deals with a flux-limited chemotaxis-haptotaxis system with nonlinear diffusion

$ \begin{eqnarray*} \left\{ \begin{split}{} &u_t = \nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(u|\nabla v|^{p-2}\nabla v)-\xi\nabla\cdot(u\nabla w)+\mu u(1-u-w), \\ &v_t = \Delta v-v+u, \\ &w_t = -vw, \ \end{split} \right. \end{eqnarray*} $

in $ \Omega\times(0, \infty) $, where $ \Omega\subseteq \mathbb{R}^{n}(n\geq2) $ is a smoothly bounded domain, $ \chi $, $ \xi $ and $ \mu $ are positive parameters, $ D(u)\geq (u+1)^{-\alpha} $ with $ \frac{2-n}{2n}<\alpha<\frac{1}{n} $. It is shown that for sufficiently smooth nonnegative initial data $ (u_{0}, v_{0}, w_{0}) $ and $ 1<p<\frac{n}{n-1}(1-\alpha) $, the corresponding initial-boundary problem possesses a unique nonnegative global classical solution, which is uniformly bounded in time.

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Mathematics Subject Classification: Primary: 92C17, 35K55, 35B35; Secondary: 35B40.

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