Convergence to equilibria of global solutions to a degenerate quasilinear Keller–Segel system

Abstract

This paper is devoted to the longtime behavior of global weak solutions to initial-boundary value problems of the following degenerate quasilinear Keller–Segel system $$\begin{aligned} {\left\{ \begin{array}{ll} u_t=\Delta u^m-\nabla \cdot (u\nabla v), &{}\quad x\in \Omega ,\;t>0\\ v_t=\Delta v-v+\gamma u, &{}\quad x\in \Omega ,\;t>0. \end{array}\right. }\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad (0.1) \end{aligned}$$ Here $$\Omega \subset \mathbb {R}^n$$ is a bounded domain, $$\gamma >0$$ and $$m>\max \{1,2-\frac{2}{n}\}$$ . First, we establish the existence of global weak solution which additionally satisfies an energy dissipation inequality. Thanks to the energy inequality, we prove that the global weak solution will converge to an equilibrium as time goes to infinity if $$m\ge 2$$ and $$\gamma <\frac{m}{m-1} M_0^{m-2}$$ , where $$M_0$$ is the average mass of cells. The proof is based on an application of a slightly modified Lojasiewicz–Simon inequality of non-smooth type, where the requirement on compactness of the trajectory is weakened compared with those in Feireisl et al. (J Differ Equ 236:551–569, 2007), Jiang and Zhang (Asymptot Anal 65:79–102, 2009). Moreover, in the special case $$m=2$$ and $$\gamma <2$$ , convergence toward trivial solution $$(M_0,\gamma M_0)$$ is verified by an alternative straightforward way.