For \(0<T\leq \infty\), and \(\Omega \subset R^ n\) with boundary in \(C^ 2\), let \(C_ T\) be the cylinder \(\Omega\) \(\times (0,T)\). The authors study the equation \[ u_ t-div[a(u,| \nabla u|)\nabla u]=c(u,| \nabla u|) \] in \(C_ T\), a, and c are real valued functions satisfying suitable conditions of smoothness. They prove that under certain stated conditions \(\Omega\) is a ball and for every \(t\in (0,T)\) u(,t) is a non-decreasing radically symmetric function with respect to the center of the ball.