Global existence and gradient estimates for a quasilinear parabolic equation of the mean curvature type with a strong perturbation

Abstract

We prove a global existence and gradient estimates of solutions to the initial-boundary problem of the quasilinear parabolic equation \[ u_t - \mbox{div}\{\sigma(|\nabla u|)^2 \nabla u\} + g(\nabla u) =0 \mbox{ in } \Omega \times (0,\infty),\] with the initial and boundary conditions $ u(0,x)=u_0(x), u|_{\partial\Omega}=0$, where $ \Omega $ is a bounded domain in $ R^N , \sigma(v)$ is a function like $ \sigma(v)=1/\!\sqrt{1+v}$ and $g(\nabla u)$ is a nonlinear perturbation like $ g(\nabla u)\!=\pm|\nabla u|^{\alpha +1}\!,$ $ \alpha >0$. In particular, we derive the estimate \[ ||\nabla u(t)||_{\infty} \leq C(||\nabla u_0||_{p_0})t^{-N/(2p_0-3N)}e^{-\lambda t}, t >0\] for a certain $\lambda > 0 $, under the assumptions that $||\nabla u_0||_{p_0}, p_0 >3(N+\alpha) \quad ( p_0 > 2\alpha+5 \mbox{ if } N=1) $, is small and the mean curvature of the boundary $ \partial \Omega $ is nonpositive.