Application of weak stabilization theory for degenerate parabolic equations in divergence form to a chemotaxis model for tumor invasion

Abstract

This paper deals with the chemotaxis model for tumor invasion, \begin{document}$ \begin{align*} \begin{cases} u_t=\nabla\cdot(D(u,w)\nabla u -u\nabla v), &x\in\Omega,\ t>0,\\ v_t=\Delta v+wz,&x\in\Omega,\ t>0,\\ w_t=-wz,&x\in\Omega,\ t>0,\\ z_t=\Delta z-z+u,&x\in\Omega,\ t>0 \end{cases} \end{align*} $\end{document} under the conditions that $ (D(u, w)\nabla u -u\nabla v)\cdot \nu = \nabla v\cdot\nu = \nabla z\cdot \nu = 0 $ on $ \partial\Omega $ and that $ (u, v, w, z)|_{t = 0} = (u_0, v_0, w_0, z_0) $, where $ \Omega \subset \mathbb{R}^N $ ($ N = 1, 2, 3 $) is a bounded domain with smooth boundary $ \partial \Omega $ and $ \nu $ is the outward normal vector to $ \partial\Omega $. The function $ D $ is supposed to satisfy $ D(0, \sigma _2) = 0 $ and $ D(\sigma _1, \sigma _2)\geq \widetilde{D}(\sigma _1) $, where $ \widetilde{D}(\sigma _1) = k\sigma _1^{m-1} $ $ (\sigma _1\leq \sigma _0) $, $ \widetilde{D}(\sigma _1) = k\sigma _0^{m-1} $ $ (\sigma _1\geq \sigma _0) $ for some $ k>0 $, $ m>1 $ and $ \sigma_0>0 $, which means that the first equation is a degenerate parabolic equation. The previous paper [4] with Fujie and Ito proved global existence and boundedness of weak solutions to the same system, however, an imperfect stabilization property was established. The purpose of this paper is to apply weak stabilization theory in [8] to obtain that$ u(\cdot, t) \to \overline{u_0}\quad \text{weakly}^*\ \text{in}\ L^\infty(\Omega) \quad \text{as}\ t \to \infty, $where $ \overline{u_0}: = \frac{1}{|\Omega|}\int_\Omega u_0 $.