Blow-up Prevention by Logistic Damping in a Chemotaxis-May-Nowak Model for Virus Infection

期刊论文
发表日期：

2024年5月8日

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Yan Li,
Qingshan Zhang

摘要

In this paper, we study the no-flux boundary initial-boundary problem for a three-component reaction-diffusion system originating from the classical May-Nowak model for viral infection $\begin{aligned} {\left\{ \begin{array}{ll} u_t=\Delta u-\chi \nabla \cdot (u\nabla v)+\kappa -u-uw-\mu u^{\alpha },\\ v_t=\Delta v-v+uw,\\ w_t=\Delta w-w+v \end{array}\right. } \end{aligned}$ $\begin{matrix} \{\begin{matrix} u_{t} = Δ u - χ \nabla \cdot (u \nabla v) + κ - u - u w - μ u^{α}, \\ v_{t} = Δ v - v + u w, \\ w_{t} = Δ w - w + v \end{matrix}) \end{matrix}$ in a smoothly bounded domain $\Omega \subset {\mathbb {R}}^n$ $Ω \subset R^{n}$ , $n\ge 1$ $n \geq 1$ . It is shown that for any $\kappa >0$ $κ > 0$ , $\mu >0$ $μ > 0$ and sufficiently regular nonnegative initial data $(u_0,v_0,w_0)$ $(u_{0}, v_{0}, w_{0})$ , the system possesses a unique nonnegative global bounded classical solution provided $\begin{aligned} \alpha >\frac{n+2}{2}. \end{aligned}$ $\begin{matrix} α > \frac{n + 2}{2} . \end{matrix}$ Moreover, we show the large time behavior of the solution with respect to the size of $\kappa $ $κ$ . More precisely, we prove that

if $\kappa <1+\mu $ $κ < 1 + μ$ , there exists $\chi _1^*$ $χ_{1}^{*}$ such that if $|\chi |\le \chi _1^*$ $| χ | \leq χ_{1}^{*}$ , then the solution satisfies $\begin{aligned} u(\cdot , t)\rightarrow u_*,\ v(\cdot , t)\rightarrow 0\ \text{ and }\ w(\cdot , t)\rightarrow 0\quad \text{ as }\ t\rightarrow \infty \end{aligned}$ $\begin{matrix} u (\cdot, t) \to u_{*}, v (\cdot, t) \to 0 and w (\cdot, t) \to 0 as t \to \infty \end{matrix}$ in $L^{\infty }(\Omega )$ $L^{\infty} (Ω)$ exponentially, where $u_*$ $u_{*}$ is the solution of algebraic equation $\begin{aligned} \kappa -y-\mu y^{\alpha }=0; \end{aligned}$ $\begin{matrix} κ - y - μ y^{α} = 0; \end{matrix}$

if $\kappa >1+\mu $ $κ > 1 + μ$ , then there exists $\chi _2^*$ $χ_{2}^{*}$ with the property that if $|\chi |\le \chi _2^*$ $| χ | \leq χ_{2}^{*}$ , then the solution fulfills that $\begin{aligned} u(\cdot , t)\rightarrow 1,\ v(\cdot , t)\rightarrow \kappa -1-\mu \ \text{ and }\ w(\cdot , t)\rightarrow \kappa -1-\mu \quad \text{ as }\ t\rightarrow \infty \end{aligned}$ $\begin{matrix} u (\cdot, t) \to 1, v (\cdot, t) \to κ - 1 - μ and w (\cdot, t) \to κ - 1 - μ as t \to \infty \end{matrix}$ in $L^{\infty }(\Omega )$ $L^{\infty} (Ω)$ .