Regularity of the free boundary for a parabolic cooperative system

TL;DR

This paper investigates the regularity of the free boundary in a parabolic cooperative system with sublinear growth, establishing optimal growth rates and proving $C^{1, eta}$ regularity in space and half-Lipschitz continuity in time near well-behaved free boundary points.

Contribution

It provides the first regularity results for the free boundary in a parabolic system with sublinear growth, including optimal growth rates and smoothness properties.

Findings

Optimal growth rate near free boundary points

$C^{1, eta}$ regularity of free boundary in space

Half-Lipschitz continuity in time

Abstract

In this paper we study the following parabolic system \begin{equation*} \Delta \u -\partial_t \u =|\u|^{q-1}\u\,\chi_{\{ |\u|>0 \}}, \qquad \u = (u^1, \cdots , u^m) \ , \end{equation*} with free boundary $\partial \{|\u | >0\}$. For $0\leq q<1$, 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.