Global Existence,Extinction And Non-extinction Of Solutions To A Class Of Fast Diffusion P-Laplace Equations

In this paper,we consider global existence,extinction and non-extinction of solutions to a class of fast diffusion p-Laplace equations:where ?pu=div(|(?)u|p-2(?)u),? is a smooth bounded domain of RN with N?2.The parameters p,q,s satisfy 1<p<2,q>0,0?s<p,and u0?(?)0,u0?L?(?)? W01,p(?).We first recall the background and recent development on the study of fast diffusion p-Laplace equations.Then we present some necessary preliminaries.Finally,we show that the solutions vanish in finite time,when q>p-1 and when u0(x)is suitably small by using Hardy-Littlewood-Sobolev inequality and comparison principle.When q?p-1,the solutions do not vanish in any finite time for negative initial energy.The main results of this paper are as follows:Theorem 1.Let u0? L?(?)? W01,p(?).If 0<q?1,then the weak solution of(0.1)exists globally;if q>1 and(?)(x),then the weak solution of(0.1)also exists globally.Here ?1>0 is the first eigenvalue of p-Laplacian in ? under homogeneous Dirichlet bound-ary condition and ?1(x)? 0 satisfying ??1??=1 is the corresponding eigenfunction.Theorem 2.Let u0?L?(?)? W01,p(?),0 ?s<p.Assume that u=u(x,t)is a weak solution of(0.1).(1)if p-1<q?1 and u0 satisfies then u(x,t)vanishes in finite time;(2)If q>1,u0 satisfies(?)and then u{x,t)vanishes in finite time.Here C1 is a positive constant depending on p,s,N(?)is the sur-face area of the unit sphere(?)B(0,1)and ? is the usual Gamma-function.Theorem 3.Let u0? L?(?)?nW01,p(?),0?s<p.Assume that u=u(x,t)is a weak solution of(0.1).(1)If q=p-1 and E(u0)?0,then u(x,t)can not vanish in any finite time;(2)If q<p-1 and E(u0)<0,then u(x,t)can not vanish in any finite time.Here(?).