The 3-Dimensional Oscillon Equation

Di Plinio, Francesco ; Duane, Gregory S. ; Temam, Roger

Bollettino della Unione matematica italiana, Série 9, Tome 5 (2012) no. 1, pp. 19-53

Résumé

On a bounded smooth domain $\Omega \subset \mathbb{R}^{3}$, we consider the generalized oscillon equation \begin{equation*}\partial_{tt} u(x, t) + \omega(t)\partial_{t}u(x, t) - \mu(t)\Delta u(x, t) + V'(u(x, t)) = 0, \qquad x \in \Omega \subset \mathbb{R}^{3}, \ t \in \mathbb{R}\end{equation*} with Dirichlet boundary conditions, where $\omega$ is a time-dependent damping, $\mu$ is a time-dependent squared speed of propagation, and $V$ is a nonlinear potential of critical growth. Under structural assumptions on $\omega$ and $\mu$ we establish the existence of a pullback global attractor $\mathcal{A} = \mathcal{A}(t)$ in the sense of [1]. Under additional assumptions on $\mu$, which include the relevant physical cases, we obtain optimal regularity of the pull-back global attractor and finite-dimensionality of the kernel sections.