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

TL;DR

This paper proves the global existence, uniqueness, and long-term convergence of solutions for a chemotaxis system with density-dependent motility and logistic growth, under specific conditions on the motility function.

Contribution

It establishes the first rigorous results on global solutions and their asymptotic behavior for this class of chemotaxis models with complex motility functions.

Findings

Solutions exist globally and are unique.

Solutions converge to the steady state as time approaches infinity.

The convergence is uniform in the domain.

Abstract

We consider a parabolic-elliptic system of partial differential equations with chemotaxis and logistic growth given by the system $$ \left\{ \begin{array}{l} u_t -\Delta (u \gamma(v)= \mu u(1-u), \\ - \Delta v +v=u, \end{array} \right. $$ under Neumann boundary conditions and appropriate initial data in a bounded and regular domain $\Omega$ of $\R^N$ (for $N \geq 1)$, where $\gamma \in C^3([0, \infty))$ and satisfies the assumptions $\gamma (s) > 0$, $\gamma^{\prime}(s) \leq 0$, $\gamma^{\prime \prime} (s) \geq 0$, $\gamma^{\prime \prime \prime}(s) \leq 0$ for any $s \geq 0$ $$-2 \gamma^{\prime}(s) + \gamma^{\prime \prime}(s)s \leq \mu_0< \mu$$ $$\frac{[\gamma^{\prime}(s)]^2}{\gamma(s)} \leq c, \quad \mbox{ for any } s \in [0, \infty). $$ We obtain the global existence and uniqueness of bounded in time solutions and the following asymptotic behavior $$\|u- 1\|_{L^{\infty}(\Omega)} +\|v- 1\|_{L^{\infty}(\Omega)} \rightarrow 0, \quad \mbox{ when } t \rightarrow +\infty.$$