JournalszaaVol. 25 , No. 4DOI 10.4171/zaa/1303

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

  • Zhang Lingling

    Taiyuan University of Technology, China
Blow-up of Solutions for a Class of Nonlinear Parabolic Equations cover

Abstract

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)2,t)f(u(x,t))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,ga, 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,ga, b,c, f, g are power functions or exponential functions.