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 $$\{\begin{array}{l}{u}_t-\Delta (u\gamma (v)=\mu u(1-u),\\ -\Delta v+v=u,\end{array}$$ 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 the assumptions $\gamma (s)>0$, ${\gamma }^{\prime }(s)\le 0$, ${\gamma }^{\prime \prime }(s)\ge 0$, ${\gamma }^{\prime \prime \prime }(s)\le 0$ for any $s\ge 0$ $$-2{\gamma }^{\prime }(s)+{\gamma }^{\prime \prime }(s)s\le {\mu }_0<\mu$$ $$\frac{[γ^{\prime}(s)]^2}{γ(s)} \leq c, \quad \mbox{ for any } s \in [0, \infty). $$ We obtain the global existence and uniqueness of bounded in time solutions and the following asymptotic behavior $$\|u- 1\|_{L^{\infty}(Ω)} +\|v- 1\|_{L^{\infty}(Ω)} \rightarrow 0, \quad \mbox{ when } t \rightarrow +\infty.$$