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Damped wave equations with fast growing dissipative nonlinearities

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  • Let $a>0$, $\Omega\subset \R^N$ be a bounded smooth domain and $-A$ denotes the Laplace operator with Dirichlet boundary condition in $L^2(\Omega)$. We study the damped wave problem

    utt$ + a u_t + A u = f(u), \ t>0, $
    $u(0)=u_0\in H^1_0(\Omega), \ \ u_t(0)=v_0\in L^2(\Omega),$

    where $f:\R\to\R$ is a continuously differentiable function satisfying the growth condition $|f(s)-f(t)|\leq C|s-t|(1+|s|^{\rho-1}+|t|^{\rho-1})$, $1<\rho<\frac{N+2}{N-2}$, ($N\geq 3$), and the dissipativeness condition $\lim$sup$_|s|\to\infty \frac{f(s)}{s}< \lambda_1$ with $\lambda_1$ being the first eigenvalue of $A$. We construct the global weak solutions of this problem as the limits as $\eta\to0^+$ of the solutions of wave equations involving the strong damping term $2\eta A^{1/2} u$ with $\eta>0$. We define a subclass $\mathcal LS\subset C([0,\infty),L^2(\Omega)\times H^{-1}(\Omega))\cap L^\infty([0,\infty),H^1_0(\Omega)\times L^2(\Omega))$ of the 'limit' solutions such that through each initial condition from $H^1_0(\Omega)\times L^2(\Omega)$ passes at least one solution of the class $\mathcal LS$. We show that the class $\mathcal LS$ has bounded dissipativeness property in $H^1_0(\Omega)\times L^2(\Omega)$ and we construct a closed bounded invariant subset A of $H^1_0(\Omega)\times L^2(\Omega)$, which is weakly compact in $H^1_0(\Omega)\times L^2(\Omega)$ and compact in $H^s_{\I}(\Omega)\times H^{s-1}(\Omega)$, $s\in[0,1)$. Furthermore A attracts bounded subsets of $H^1_0(\Omega)\times L^2(\Omega)$ in $H^s_\{I\}(\Omega)\times H^{s-1}(\Omega)$, for each $s\in[0,1)$. For $N=3,4,5$ we also prove a local uniqueness result for the case of smooth initial data.

    Mathematics Subject Classification: Primary: 35L05; Secondary: 35B25, 35B33, 35B40.


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