Regularity of the free boundary for a parabolic cooperative system

Abstract

In this paper we study the following parabolic system \begin{equation*} Δ\u -\partial_t \u =|\u|^{q-1}\u\,χ_{\{ |\u|>0 \}}, \qquad \u = (u^1, \cdots , u^m) \ , \end{equation*} with free boundary $\partial \{|\breve{|}>0\}$. For $0\le q<1$, we prove optimal growth rate for solutions $\u $ to the above system near free boundary points, and show that in a uniform neighbourhood of any a priori well-behaved free boundary point the free boundary is $C^{1,\alpha }$ in space directions and half-Lipschitz in the time direction.