The 3-Dimensional Oscillon Equation

Di Plinio, Francesco, Duane, Gregory S., and Temam, Roger. "The 3-Dimensional Oscillon Equation." Bollettino dell'Unione Matematica Italiana 5.1 (2012): 19-53. <http://eudml.org/doc/290848>.

@article{DiPlinio2012,
abstract = {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.},
author = {Di Plinio, Francesco, Duane, Gregory S., Temam, Roger},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {19-53},
publisher = {Unione Matematica Italiana},
title = {The 3-Dimensional Oscillon Equation},
url = {http://eudml.org/doc/290848},
volume = {5},
year = {2012},
}

TY - JOUR
AU - Di Plinio, Francesco
AU - Duane, Gregory S.
AU - Temam, Roger
TI - The 3-Dimensional Oscillon Equation
JO - Bollettino dell'Unione Matematica Italiana
DA - 2012/2//
PB - Unione Matematica Italiana
VL - 5
IS - 1
SP - 19
EP - 53
AB - 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.
LA - eng
UR - http://eudml.org/doc/290848
ER -