# American Institute of Mathematical Sciences

November  2002, 8(4): 939-951. doi: 10.3934/dcds.2002.8.939

## Estimates on the dimension of a global attractor for a semilinear dissipative wave equation on $\mathbb R^N$

 1 Department of Statistics and Actuarial Science, University of the Aegean, Karlovassi 83200, Samos, Greece 2 Department of Mathematics, National Technical University, Zografos Campus 15780, Athens, Greece

Received  April 2001 Revised  May 2002 Published  July 2002

We discuss estimates of the Hausdorff and fractal dimension of a global attractor for the semilinear wave equation

$u_{t t} +\delta u_t -\phi (x)\Delta u + \lambda f(u) = \eta (x), x \in \mathbb R^N, t \geq 0,$

with the initial conditions $u(x,0) = u_0 (x)$ and $u_t(x,0) = u_1 (x),$ where $N \geq 3$, $\delta >0$ and $(\phi (x))^{-1}:=g(x)$ lies in $L^{N/2}(\mathbb R^N)\cap L^\infty (\mathbb R^N)$. The energy space $\mathcal X_0=\mathcal D^{1,2}(\mathbb R^N) \times L_g^2(\mathbb R^N)$ is introduced, to overcome the difficulties related with the non-compactness of operators, which arise in unbounded domains. The estimates on the Hausdorff dimension are in terms of given parameters, due to an asymptotic estimate for the eigenvalues $\mu$ of the eigenvalue problem $-\phi(x)\Delta u=\mu u, x \in \mathbb R^N$.

Citation: Nikos I. Karachalios, Nikos M. Stavrakakis. Estimates on the dimension of a global attractor for a semilinear dissipative wave equation on $\mathbb R^N$. Discrete & Continuous Dynamical Systems, 2002, 8 (4) : 939-951. doi: 10.3934/dcds.2002.8.939
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