Blow-up,Non-extinction And Extinction Of The Solution To A Parabolic P-Laplace Equation

We study blow-up and non-extinction of the solution to a parabolic p-Laplace equation ut-div(|(?)u|p-2(?)u)= |u|q-2ulog |u|-fΩ|u|q-2ulog |u|dx,subject to homogeneous Neumann boundary value conditions.For the case of 1<p<2,we prove that,under the condition of non-positive initial energy,the solution blows up in finite time if q>2,and the solution does not extinct in finite time if 1<q≤ p,the solution extinct in finite time if 1<p<2,<q<2.The thesis consists of two sections.Chapter 1 is the preface.In Chapter 2,we consider the following parabolic p-Laplace equation where Ω is a smooth bounded domain of RN,1<p<2,q/1,fΩvdx =1/|Ω|∫vdx,and u0 satisfies u0∈L∞(Ω)∩W1,p(Ω)\{0},∫Ωu0dx=0,(2)Then we have the following theorems.Theorem 2.1.1.If u is the solution of(3),with 1<p<2,q>2,E(u0)≤0 and u0 satisfies condition(4),then the solution of problem(3)blows up in finite time.Theorem 2.1.2.If u is the solution of(3),with 1<p<2,1<q≤p and u0 satisfies condition(4),then the solution of problem(3)does not extinct.Theorem 2.1.3.If u is the solution of(3),with 1<p<2,p<q<2 and u0 satisfies condition(4),then the solution of problem(3)extinct in finite time providedΩ|2(2-q-α)/ps+s‖u0‖ps+2 q+2-p+|Ω|2(2-q)/ps+s‖u0‖ps+2 q-2-p<eαC0,where C0=ps+1/(s+1)pC*-p|Ω|p/p*-p(s=1)/ps+2,and s ≥ 0,ps+2/s+1≤p*=np/n-p.