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

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  • 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 $W_0^{1,p}(\Omega)$.

    Mathematics Subject Classification: 37C45.


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