Global and blowup solutions of quasilinear parabolic equation with critical Sobolev exponent and lower energy initial value

Abstract

In this paper, by means ofthe energy method, we first study the existence and asymptotic estimates of global solution of quasilinear parabolic equations involving p-Laplacian (p> 2)and critical Sobolevexponentand lowerenergy initialvalue inaboundeddomain in RV(N> 3),andalsostudythesufficientconditionsoffinitetimeblowupoflocalsolutionby the classical concave method. Finally, we study the asymptotic behavior of any global solutions u(x, t; Uo)whichmaypossesshighenergy initialvalue function Uo(X).Wecanprove that there exists a time subsequence {t,} such that the asymptotic behavior of u(x, t,,; Uo) as t, o is similar to the Palais-Smale sequence of stationary equation of the above parabolic problem.