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Zeitschrift für Analysis und ihre Anwendungen

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Volume 25, Issue 4, 2006, pp. 479–486
DOI: 10.4171/ZAA/1303

Published online: 2006-12-31

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

Zhang Lingling[1]

(1) Taiyuan University of Technology, China

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.

Keywords: Nonlinear parabolic equations, blow-up solutions, maximum principles

Lingling Zhang: Blow-up of Solutions for a Class of Nonlinear Parabolic Equations. Z. Anal. Anwend. 25 (2006), 479-486. doi: 10.4171/ZAA/1303