Finite speed of propagation in 1-D degenerate Keller-Segel system

Abstract

We consider the following Keller-Segel system of degenerate type: \partial u / \partial t = \partial / \partial x (\partial u^m / \partial x - u^{q-1} \cdot \partial v / \partial x), x \in \R, t>0, \partial^2 v / \partial x^2 - \gamma v + u, x \in \R, t>0, u(x,0) = u_0(x), x \in \R, where $m>1, \gamma > 0, q \ge 2m$. We shall first construct a weak solution $u(x,t)$ of (KS) such that $u^{m-1}$ is Lipschitz continuous and such that $\displaystyle u^{m-1+\delta}$ for $\delta>0$ is of class $C^1$ with respect to the space variable $x$. As a by-product, we prove the property of finite speed of propagation of a weak solution $u(x,t)$ of (KS), {\it i.e.,} that a weak solution $u(x,t)$ of (KS) has a compact support in $x$ for all $t>0$ if the initial data $u_0(x)$ has a compact support in $\R$. We also give both upper and lower bounds of the interface of the weak solution $u$ of (KS).