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

TL;DR

This paper studies the persistence and long-term behavior of solutions to a chemotaxis system with logistic growth on ^N, proving global existence, boundedness, and convergence under certain parameter conditions.

Contribution

It establishes conditions for global existence, persistence, and convergence of solutions to a chemotaxis system with logistic source, extending understanding of its long-term dynamics.

Findings

Global existence of solutions under specific parameter conditions.

Persistence of solutions with positive lower bounds.

Asymptotic convergence to steady states when parameters satisfy certain inequalities.

Abstract

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}\,\,\, t>0\cr {v_t}=\Delta v -\lambda v+\mu u,\quad x\in \mathbb{R}^{N}\,\,\, t>0 \end{cases}(1) \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 $$\lim_{t\to\infty}\big[\|u(x,t;u_0, v_0)-\frac{a}{b}\|_{\infty}+\|v(x,t;u_0, v_0)-\frac{\mu}{\lambda}\frac{a}{b}\|_{\infty}\big]=0.$$