Journal of Inequalities and Applications 
Volume 6 (2001), Issue 1, Pages 57-75
doi:10.1155/S1025583401000054

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

Zhong Tan1 and Zheng-An Yao2

 1Department of Mathematics, Xiamen University, Fujian, Xiamen 361005, China
2Department of Mathematics, Zhongshan University, Guangdong, Guangzhou 510275, China
Received 25 August 1999; Revised 22 October 1999

Abstract

In this paper, by means of the energy method, we first study the existence and asymptotic estimates of global solution of quasilinear parabolic equations involving p-Laplacian (p>2) and critical Sobolev exponent and lower energy initial value in a bounded domain in RN(N≥3), and also study the sufficient conditions of finite time blowup of local solution by the classical concave method. Finally, we study the asymptotic behavior of any global solutions u(x,t;u0) which may possess high energy initial value function u0(x). We can prove that there exists a time subsequence {tn} such that the asymptotic behavior of u(x,tn;u0) as tn→∞ is similar to the Palais–Smale sequence of stationary equation of the above parabolic problem.