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

We consider the following Keller‐Segel system of degenerate type: (KS): \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\frac{\partial u}{\partial t} = \frac{\partial }{\partial x} \big ( \frac{\partial u^m}{\partial x} - u^{q-1} \frac{\partial v}{\partial x} \big ), x \in {\mathbb R}, t>0, 0 = \frac{\partial ^2 v}{\partial x^2} - \gamma v + u, x \in {\mathbb R}, t>0, u(x,0) = u_0(x), x \in {\mathbb R},$\end{document} where m > 1, γ > 0,  q ⩾ 2 m . We shall first construct a weak solution u ( x , t ) of (KS) such that u m − 1 is Lipschitz continuous and such that \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\displaystyle u^{m-1+\delta }$\end{document} for δ > 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), 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 \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathbb R}$\end{document} . We also give both upper and lower bounds of the interface of the weak solution u of (KS).