Blow-up Of Solutions To A Class Of Nonlinear Parabolic Equations With Special Diffusion Coefficients

In this paper,we consider the blow-up properties of solutions of a class of nonlinear parabolic equations with special diffusion coefficients where Ω(?)Rn(n≥ 3)is a bounded domain with a smooth boundary(?)Ω,0∈Ω,△pu=div(|▽u|p-2▽u),2<p<n,0≤u0∈W01,p(Ω),u0(x)(?)0.In addition,forx=(x1,x2,…,xn)∈Rn,|x|=(?).The weight function k(t)and the nonlinear function f satisfy the following assumptions:(ⅰ)k ∈ C1[0,+∞),k(0)>0,k(t)≥ 0,■t ∈[0,+∞);(ⅱ)sf(s)≥0,(?)s ∈ R;(ⅲ)f ∈ C1(R)and there exists a constant q>1 such that sf(s)≥(q+1)F(s),(?)s∈R;(ⅳ)There exist constants a,b>0 and q ∈(1,p*-1)such that|f(s)| ≤ a+b|s|q,(?)s ∈ R.Here p*=np/n-p for n>p and p*=∞ for n≤p.At first,the research background and current research status of nonlinear parabolic e-solution to problem(0.1)that blows up at T.Then T≥:L1-γ(0)/C*(γ-1),where L(0)=1/2‖u0/|x|‖22,γ>1 and C*>0 are constants depending on n,p and q.