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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