Abstract: In this paper, we investigate the following partial differential equation, u_t-a(x·▽u)|u|^p-1=△u+|u|^p-1u, where a ≥ 0 and p> 1. When n(p-1)/2 > 1 and p > 3, we obtained a nontrivial non-negative global solution. Furthermore, on Sobolev space W^1,s(W^2,s) with s > 1. a nonuniqueness result is established which shows that there exists a positive solution u(t,x) with u(t,x)→0 as t→0 in W^1,s(W^2,s). On the other hand, our result can be regarded as a generalization of conclusion of Haraux, A.and Weissler, F.B. in[5].

Key words: Existence and asymptotic behavior, self-similar solution, non-uniqueness