Blow-up of Solutions for a Class of Nonlinear Parabolic Equations

Abstract

In this paper, the blow up of solutions for a class of nonlinear parabolic equations u_t(x,t)=\nabla _{x}(a(u(x,t))b(x)c(t)\nabla _{x}u(x,t))+g(x,|\nabla _{x}u(x,t) |^2,t)f(u(x,t)) with mixed boundary conditions is studied. By constructing an auxiliary function and using Hopf's maximum principles, an existence theorem of blow-up solutions, upper bound of “blow-up time” and upper estimates of “blow-up rate” are given under suitable assumptions on a, b,c, f, g , initial data and suitable mixed boundary conditions. The obtained result is illustrated through an example in which a, b,c, f, g are power functions or exponential functions.