September  2004, 3(3): 475-490. doi: 10.3934/cpaa.2004.3.475

The existence and asymptotic behavior of the evolution p-Laplacian equations with strong nonlinear sources

1. 

Department of Mathematics, University of Xiamen, Fujian Xiamen 361005, China

2. 

Department of Mathematics, University of Zhongshan, Guangzhou 510275, China

Received  December 2002 Revised  April 2004 Published  June 2004

In this paper we consider the existence, nonexistence and the asymptotic behavior of the global solutions of the quasilinear parabolic equation of the following form:

$u_t-\Delta_pu=|u|^{q-2}u, \quad (x,t)\in\Omega\times (0,T),$

$u(x,t)=0,\quad (x,t)\in\partial\Omega\times (0,T), $

$ u(x,0)=u_0(x), \quad u_0(x)\geq 0, u_0(x)$ ≠ $0, $

where $\Omega$ is a smooth bounded domain in $R^N(N\geq 3)$, $\Delta_pu=$ div$(|\nabla u|^{p-2}\nabla u )$, $\frac{2N}{N+2}$ < $p$ < $N$, $q=p^\star=\frac{pN}{N-p}$ is the critical Sobolev exponent. In particular, we employ the concentration-compactness principle to prove that the global solutions with the initial data in "stable set" converge strongly to zero in $W_0^{1,p}(\Omega)$.

Citation: Zhong Tan, Zheng-An Yao. The existence and asymptotic behavior of the evolution p-Laplacian equations with strong nonlinear sources. Communications on Pure and Applied Analysis, 2004, 3 (3) : 475-490. doi: 10.3934/cpaa.2004.3.475
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