Global well-posedness and flat-hump-shaped stationary solutions for degenerate chemotaxis systems with threshold density

Abstract

In a smoothly bounded domain $\Omega \subset \mathbb{R}^N$ $(N\in \mathbb{N})$, a no-flux initial-boundary value problem for the degenerate chemotaxis system with volume-filling effects, \begin{align*} u_t = \nabla \cdot (D(u,v) \nabla u - h(u,v) \nabla v), \quad v_t = \Delta v + g(u,v), \quad x\in \Omega, \ t>0, \end{align*} is considered under the assumptions that $D(1,s)=0$ and that $h(0,s)=h(1,s)=0$. Here, initial data $u_0$ and $v_0$ have suitable regularity and satisfy $0\le u_0\le 1$ and $v_0\ge 0$ with $\nabla v_0 \cdot \nu|_{\partial \Omega} = 0$. It is proved that there exists a global weak solution such that $0\le u\le 1$ and $v\ge 0$. Moreover, when $D(r,s) = D(r)$ for all $r\in[0,1]$ and $s\in[0,\infty)$ and additional conditions on $D$, $h$ and $g$ are assumed, uniqueness of global weak solutions with the mass conservation law $\int_\Omega u(x,t) \, dx = \int_\Omega u_0(x) \, dx$ is shown. Also, a flat-hump-shaped stationary solution is constructed in the one-dimensional setting