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

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

$$\begin{aligned} {\left\{ \begin{array}{ll} u_t=\Delta u-\chi \nabla \cdot (u\nabla v)+\kappa -u-uw-\mu u^{\alpha },\\ v_t=\Delta v-v+uw,\\ w_t=\Delta w-w+v \end{array}\right. } \end{aligned}$$

in a smoothly bounded domain \(\Omega \subset {\mathbb {R}}^n\), \(n\ge 1\). It is shown that for any \(\kappa >0\), \(\mu >0\) and sufficiently regular nonnegative initial data \((u_0,v_0,w_0)\), the system possesses a unique nonnegative global bounded classical solution provided

$$\begin{aligned} \alpha >\frac{n+2}{2}. \end{aligned}$$

Moreover, we show the large time behavior of the solution with respect to the size of \(\kappa \). More precisely, we prove that

• if \(\kappa <1+\mu \), there exists \(\chi _1^*\) such that if \(|\chi |\le \chi _1^*\), then the solution satisfies

  $$\begin{aligned} u(\cdot , t)\rightarrow u_*,\ v(\cdot , t)\rightarrow 0\ \text{ and }\ w(\cdot , t)\rightarrow 0\quad \text{ as }\ t\rightarrow \infty \end{aligned}$$

  in \(L^{\infty }(\Omega )\) exponentially, where \(u_*\) is the solution of algebraic equation

  $$\begin{aligned} \kappa -y-\mu y^{\alpha }=0; \end{aligned}$$

• if \(\kappa >1+\mu \), then there exists \(\chi _2^*\) with the property that if \(|\chi |\le \chi _2^*\), then the solution fulfills that

  $$\begin{aligned} u(\cdot , t)\rightarrow 1,\ v(\cdot , t)\rightarrow \kappa -1-\mu \ \text{ and }\ w(\cdot , t)\rightarrow \kappa -1-\mu \quad \text{ as }\ t\rightarrow \infty \end{aligned}$$

  in \(L^{\infty }(\Omega )\).