Blow Up Of Solutions To Some Reaction Diffusion Equations With Variable Sources

This paper mainly studies the blow-up property for the following equation with ho-mogenous Dirichlet boulldary condition or dynamical boundary condition. Where K∈N,m∈N.The blow-up property for(0.1)were studied by many mathematicians Such as F.B. Weissler,A.Friedman,B.Mcleod,J.Von Below and G.Pincet Mailly etc.We will state some of their results in the first part of this paper. In the second part we will study the blow-up property for the equation defined by(0.1)with A(x,u,▽u)=▽u,F0(x,t,u,▽u)=|u|p(x)-1u under Dirichlet boundary condition B(x,u,▽u)=u=0or the dynamical boundary condition B(x,u,▽u)=u2k(?)tu+|▽u|m-2(?)vu-|u|p(x)-1u=0. Denote p+=sup∈Ωp(x), p-=infx∈Ω p(x).The main results of this paper are the following.(1)Let B(x,u,▽u)=u=0.Suppose that u is a weak solution of the problem,define E1(t)=1/m∫Ω｜▽u｜mdx-∫Ω1/p(x)+1dx,and N(t)=∫Ωu2k+2dx.Ifφ0>0,and E1(0)≤0, then the week solution of(0.1)will blow up in a finite time T which satisfies T≤t1=N(0)1-α/M1(α-1) where α＝p-+1/2+2>1,M1is a constant depending only on|Ω|,p+,p-,N(0) and k.(2)Let B(x,u,▽u)=u2k(?)tu+|▽u|m-2(?)vu-|u|p(x)-1u=0.Define E2(t)=1/m∫Ω｜▽u|mdx-(∫Ω1/p(x)+1|u|p(x)+1dx+∫(?)Ω1/P(x)+1|u|p(x)+1ds) and G(t)=∫Ω2k+2dx+1/2k+1∫(?)Ω u2k+2ds.If φ0>0anand E2(0)≤0,then the week solution of (0.1)will blow up in a finite time T with T≤t2=G(0)1-α/M2(α-1) where α＝p-+1/2k+2>1,M2is a constant depending only on|Ω|,p-,p+,G(0) and k.