Qualitative behavior of solutions for a chemotaxis-haptotaxis system with gradient-dependent flux-limitation

Abstract

This paper deals with the coupled chemotaxis-haptotaxis model with gradient-dependent flux-limitation $\{\begin{array}{l}{u}_t=\mathrm{\Delta }u-\chi \mathrm{\nabla }\cdot (u|\mathrm{\nabla }v{|}^{p-2}\mathrm{\nabla }v)-\xi \mathrm{\nabla }\cdot (u\mathrm{\nabla }w)\\ +\mu u(1-u-w),& x\in \mathrm{\Omega },t>0,\\ 0=\mathrm{\Delta }v-v+u,& x\in \mathrm{\Omega },t>0\\ w_t=-vw,& x\in \mathrm{\Omega },t>0,\end{array}$ where $\mathrm{\Omega }\subset {\mathbb{R}}^n(n\ge 1)$ is a bounded domain with smooth boundary, χ, ξ and μ are positive parameters.

• Under the condition that $1<p<\frac{n}{n-1},$ then for any $\chi ,\xi ,\mu >0$ and all reasonably smooth initial data $(u_0,w_0)$, the above system possesses a globally classical solution, which is uniformly bounded in time.

• Under the conditions that $1<p<\frac{n}{n-1}\mathrm{and}\mu >\frac{{\chi }^2}{8},$

it asserts exponential decay of w in the large time limit, whereas both u and v persist in some certain sense.

Keywords:

• Boundedness
• asymptotic properties
• chemotaxis-haptotaxis
• flux-limitation

AMS(2020) Subject Classifications:

• 92C17
• 35K55
• 35B35
• 35B40

Acknowledgments

The authors would like to deeply thank the editor and anonymous reviewers for their insightful and constructive comments. Pan Zheng is deeply grateful to Professor Renjun Duan for the friendly hospitality in The Chinese University of Hong Kong.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work is partially supported by National Natural Science Foundation of China [grant numbers 11601053, 12271064], Natural Science Foundation of Chongqing [grant number cstc2019jcyj-msxmX0082], the Science and Technology Research Project of Chongqing Municipal Education Commission (Grant No. KJZD-K202200602), China-South Africa Young Scientist Exchange Project in 2020, The Hong Kong Scholars Program [grant numbers XJ2021042, 2021-005] and Young Hundred Talents Program of CQUPT in 2022–2024.