Symmetry in a free boundary problem for degenerate parabolic equations on unbounded domains

We use the method of Alexandroff-Serrin to establish the spherical symmetry of the ground domain and of the weak solution to a free boundary problem for a class of quasi-linear parabolic equations in an unbounded cylinder $\Omega \times (0,T)$, where $\Omega = (\mathbb{R} ^{n} \backslash \overline{\Omega_{1}})$, with $\Omega_{1}\subset \mathbb R^n$ a simply connected bounded domain. The equations considered are of the type $u_{t} - div (a(u,\vert Du\vert)Du) = c(u,\vert Du\vert)$, with $a$ modeled on $\vert Du\vert^{p-2}$. We consider a solution satisfying the boundary conditions: $u(x,t)=f(t)$ for $(x,t)\in \partial \Omega_{1} \times (O,T)$, and $u(x,0)=0$, $u\rightarrow 0$ as $\vert x\vert\rightarrow\infty$. We show that the overdetermined co-normal condition $a(u,\vert Du\vert)\frac{\partial u}{\partial\nu}=g(t)$ for $(x,t)\in \partial \Omega_{1} \times (O,T)$, 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.