Blow-Up of Solutions for a Class of Reaction-Diffusion Equations with a Gradient Term under Nonlinear Boundary Condition

Abstract

The blow-up of solutions for a class of quasilinear reaction-diffusion equations with a gradient term u t = div(a(u)b(x)∇u) + f (x, u, |∇u| 2 ,t) under nonlinear boundary condition ∂ u/∂ n + g(u) = 0 are studied. By constructing a new auxiliary function and using Hopf's maximum principles, we obtain the existence theorems of blow-up solutions, upper bound of blow-up time, and upper estimates of blow-up rate. Our result indicates that the blow-up time T * may depend on a(u), while being independent of g(u) and f .