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

In this paper, the blow up of solutions for a class of nonlinear parabolic equations ut(x, t) = ∇x(a(u(x, t))b(x)c(t)∇xu(x, t)) + g(x, |∇xu(x, t)| , 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.