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

Pages: 475 - 490, Volume 3, Issue 3, September 2004

doi:10.3934/cpaa.2004.3.475       Abstract        Full Text (243.9K)       Related Articles

Zhong Tan - Department of Mathematics, University of Xiamen, Fujian Xiamen 361005, China (email)
Zheng-An Yao - Department of Mathematics, University of Zhongshan, Guangzhou 510275, China (email)

Abstract: 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 $W_0^{1,p}(\Omega)$.

Keywords:  p-Laplace evolution equation, critical Sobolev exponent, Asymptotic behavior, Finite time blowup.
Mathematics Subject Classification:  37C45.

Received: December 2002;      Revised: April 2004;      Published: June 2004.