\(C^{1,\alpha}\) regularity of interface of some nonlinear degenerate parabolic equations (Q1586658)

The author considers the equation \(u_t=\text{div} (|\nabla u^m|)^{p-2} \nabla u^m\) in \({\mathbb R}^N \times (0,T)\) with the initial condition \(u(x,0) =u_0 (x)\). It is known that there is an interface that separates the region where \(u>0\) from the region where \(u=0\). Let \(\Omega \) be a region where \(u>0\), \(\Gamma = \partial \Omega \cap {t>0}\). The main result is concluded in the regularity of \(\Gamma \). Namely the author establishes under natural assumptions that \(\Gamma\) is presented by a \(C^{1, \alpha }\) function, i.e. a Lipschitz continuous function.