The 3-Dimensional Oscillon Equation

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

Bollettino dell'Unione Matematica Italiana, Tome 5 (2012), p. 19-53 / Harvested from Biblioteca Digitale Italiana di Matematica

Résumé

On a bounded smooth domain $\Omega \subset \mathbb{R}^{3}$, we consider the generalized oscillon equation ∂t⁢t⁡u⁢(x,t)+ω⁢(t)⁢∂t⁡u⁢(x,t)-μ⁢(t)⁢Δ⁢u⁢(x,t)+V′⁢(u⁢(x,t))=0,x∈Ω⊂ℝ3,t∈ℝ 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.

Détail

Publié le : 2012-02-01

MR 2919647 | Zbl 1256.35155

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