Blow-up of solutions for a class of quasilinear parabolic equations with nonlinear Boundary conditions

Abstract

In this paper, we study the blow-up of solutions for a class of quasilinear parabolic equations ut = div (a (u)∇u) + f (x, u, | ∇u | 2, t) with nonlinear boundary conditions ∂u/∂n +g (u) = 0. By constructing a new auxiliary function and using Hopf ’s maximum principles, we obtain the existence theorem of blow-up solutions, upper bound of time, and upper estimates of blow-up rate. Our result indicates that the blow-up time T * may depend on g (u), while be independent of f.