Weak stabilization in degenerate parabolic equations in divergence form: application to degenerate Keller–Segel systems

Abstract

We consider the initial-boundary value problem for the degenerate parabolic equation

$$\begin{aligned} u_t=\nabla \cdot (f(u)\nabla u+{g(u,x,t)}),\quad x\in \Omega ,\ t>0 \end{aligned}$$

in a smooth bounded domain $\Omega \subset \mathbb {R}^N(N\in \mathbb {N})$ under the no-flux boundary condition with a non-negative initial data $u_0\in L^\infty (\Omega )$. Here f is a non-negative function belonging to ${C([0,\infty )) \cap C^2((0,\infty ))}$, and g is a vector-valued function on $[0,\infty ) \times \Omega \times (0,\infty )$. It is known that this problem has a global-in-time weak solution by the well-known parabolic theory. This paper shows the stabilization in this problem; in detail, the problem admits a global weak solution which fulfills

$$\begin{aligned} u(t)\rightarrow \frac{1}{|\Omega |}\int _\Omega u_0 \quad \text{ weakly}^* \text{ in } L^\infty (\Omega ) \text{ as } t\rightarrow \infty . \end{aligned}$$