JournalszaaVol. 25, No. 4pp. 479–486

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

  • Zhang Lingling

    Taiyuan University of Technology, China
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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.

Cite this article

Zhang Lingling, Blow-up of Solutions for a Class of Nonlinear Parabolic Equations. Z. Anal. Anwend. 25 (2006), no. 4, pp. 479–486

DOI 10.4171/ZAA/1303