On a comparison method for a parabolic–elliptic system of chemotaxis with density-suppressed motility and logistic growth

Abstract

We consider a parabolic–elliptic system of partial differential equations with chemotaxis and logistic growth given by the system

$$\begin{aligned} \left\{ \begin{array}{l} u_t -\Delta (u \gamma (v))= \mu u(1-u), \\ - \Delta v +v=u, \end{array} \right. \end{aligned}$$

under Neumann boundary conditions and appropriate initial data in a bounded and regular domain $\Omega$ of ${{\mathbb {R}}}^N$ (for $N \ge 1)$, where $\gamma \in C^3([0, \infty ))$ and satisfies $\gamma (s) > 0$, $\gamma ^{\prime }(s) \le 0$, $\gamma ^{\prime \prime } (s) \ge 0$, $\gamma ^{\prime \prime \prime }(s) \le 0$ for any $s \ge 0$

$$\begin{aligned}{} & {} -2 \gamma ^{\prime }(s) + \gamma ^{\prime \prime }(s)s \le \mu _0< \mu \\{} & {} \frac{[\gamma ^{\prime }(s)]^2}{\gamma (s)} \le c, \quad \text{ for } \text{ any } s \in [0, \infty ). \end{aligned}$$

We obtain the global existence and uniqueness of bounded in time solutions and the following asymptotic behavior

$$\begin{aligned} \Vert u- 1\Vert _{L^{\infty }(\Omega )} +\Vert v- 1\Vert _{L^{\infty }(\Omega )} \rightarrow 0, \quad \text{ when } t \rightarrow +\infty . \end{aligned}$$