Persistence and convergence in parabolic-parabolic chemotaxis system with logistic source on <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^{N} $\end{document}</tex-math></inline-formula>

Wenxian Shen; Shuwen Xue

doi:10.3934/dcds.2022003

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June 2022, 42(6): 2893-2925. doi: 10.3934/dcds.2022003

Persistence and convergence in parabolic-parabolic chemotaxis system with logistic source on $ \mathbb{R}^{N} $

Wenxian Shen ^1, and Shuwen Xue ^1,2,,

1.	Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA
2.	Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NL A1C 5S7, Canada

^*Corresponding author: Shuwen Xue (szx0006@auburn.edu; sxue@mun.ca)

Received March 2021 Revised November 2021 Published June 2022 Early access January 2022

In the current paper, we consider the following parabolic-parabolic chemotaxis system with logistic source on

$ \mathbb{R}^{N} $

$ \begin{equation} \begin{cases} u_{t} = \Delta u - \chi\nabla\cdot ( u\nabla v) + u(a-bu),\quad x\in{{\mathbb R}}^N,\\ {v_t} = \Delta v -\lambda v+\mu u,\quad x\in{{\mathbb R}}^N,\,\,\, \end{cases} \;\;\;\;\;\;\;\;\left( 1 \right)\end{equation} $

where

$ \chi, \ a,\ b,\ \lambda,\ \mu $

are positive constants and

$ N $

is a positive integer. We investigate the persistence and convergence in (1). To this end, we first prove, under the assumption

$ b>\frac{N\chi\mu}{4} $

, the global existence of a unique classical solution

$ (u(x,t;u_0, v_0),v(x,t;u_0, v_0)) $

of (1) with

$ u(x,0;u_0, v_0) = u_0(x) $

and

$ v(x,0;u_0, v_0) = v_0(x) $

for every nonnegative, bounded, and uniformly continuous function

$ u_0(x) $

, and every nonnegative, bounded, uniformly continuous, and differentiable function

$ v_0(x) $

. Next, under the same assumption

$ b>\frac{N\chi\mu}{4} $

, we show that persistence phenomena occurs, that is, any globally defined bounded positive classical solution with strictly positive initial function

$ u_0 $

is bounded below by a positive constant independent of

$ (u_0, v_0) $

when time is large. Finally, we discuss the asymptotic behavior of the global classical solution with strictly positive initial function

$ u_0 $

. We show that there is

$ K = K(a,\lambda,N)>\frac{N}{4} $

such that if

$ b>K \chi\mu $

and

$ \lambda\geq \frac{a}{2} $

, then for every strictly positive initial function

$ u_0(\cdot) $

, it holds that

$ \lim\limits_{t\to\infty}\big[\|u(x,t;u_0, v_0)-\frac{a}{b}\|_{\infty}+\|v(x,t;u_0, v_0)-\frac{\mu}{\lambda}\frac{a}{b}\|_{\infty}\big] = 0. $

Keywords: Parabolic-parabolic chemotaxis system, logistic source, classical solution, global existence, persistence, asymptotic behavior.

Mathematics Subject Classification: 35A01, 35B35, 35B40, 35Q92, 92C17.

Citation: Wenxian Shen, Shuwen Xue. Persistence and convergence in parabolic-parabolic chemotaxis system with logistic source on $ \mathbb{R}^{N} $. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 2893-2925. doi: 10.3934/dcds.2022003