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