Symmetry in a free boundary problem for degenerate parabolic equations on unbounded domains
spherical symmetry of the ground domain and of the solution
The following problem is considered: \(u_t-\text{ div } (a(u,|Du|)Du)=c(u,|Du|)\) in \(C_T=(\mathbb R^n \setminus\overline{\Omega}_1)\times (0,T)\); \(u(x,t)=f(t)\) on \(\partial\Omega_1\times (0,T)\), \(f(0)=0\); \(u(x,0)=0\) if \(x \in \mathbb R^n\setminus\overline{\Omega}_1\); \(u(x,t)\rightarrow 0\) uniformly in \(t \in (0,T)\) as \(|x|\rightarrow\infty\); \(0\leq u\leq f\) in \(C_T\); \(\Omega_1\) is a simply-connected bounded \(C^{2,\alpha}\)-domain. Using the Alexandroff--Serrin method and some assumptions for \(a\), \(c\), \(f\), the authors showed (from their abstract) ``\dots that the overdetermined co-normal condition \(a(u,|Du|)\frac{\partial u} {\partial\nu}=g(t)\) for \((x,t) \in \partial\Omega_1\times (0,T)\), with \(g(\overline T)>0\) for at least one value \(\overline T \in (0,T)\), forces the spherical symmetry of the ground domain and of the solution''.
scientific article; zbMATH DE number 1663101
scientific article