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

keywords: Dynamical systems hyperbolic equations unbounded domains attractors Hausdorff dimension. generalized Sobolev spaces
On the dynamics of a degenerate damped semilinear wave equation in \mathbb{R}^N : the non-compact case
Nikos I. Karachalios Athanasios N Lyberopoulos
Conference Publications 2007, 2007(Special): 531-540 doi: 10.3934/proc.2007.2007.531
We show the existence of a global attractor for a degenerate, linearly damped, semilinear wave equation in $mathbb{R}^N$ under a new condition concerning a variable non-negative diffusivity. In particular, we show the asymptotic compactness of the induced semiflow by combining the energy equation method with appropriate tail estimates.
keywords: tail estimates global attractor asymptotically compact semiflow. Degenerate semilinear wave equation energy equation method

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