Fractional oscillon equations; solvability and connection with classical oscillon equations

TL;DR

This paper investigates the asymptotic behavior of nonautonomous fractional approximations of oscillon equations, establishing the existence of time-dependent attractors under certain conditions.

Contribution

It introduces a framework for analyzing fractional oscillon equations with time-dependent parameters and proves the existence of attractors in this setting.

Findings

Existence of time-dependent attractors for fractional oscillon equations.

Conditions under which attractors exist for nonautonomous models.

Connection established between fractional and classical oscillon equations.

Abstract

In this paper we are concerned with the asymptotic behavior of nonautonomous fractional approximations of oscillon equation $$ u_{tt}-\mu(t)\Delta u+\omega(t)u_t=f(u),\ x\in\Omega,\ t\in\mathbb{R}, $$ subject to Dirichlet boundary condition on $\partial \Omega$, where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$, $N\geqslant 3$, the function $\omega$ is a time-dependent damping, $\mu$ is a time-dependent squared speed of propagation, and $f$ is a nonlinear functional. Under structural assumptions on $\omega$ and $\mu$ we establish the existence of time-dependent attractor for the fractional models in the sense of Carvalho, Langa, Robinson \cite{CLR}, and Di Plinio, Duane, Temam \cite{DDT1}.