Fractional oscillon equations; solvability and connection with 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 ${\mathbb{R}}^N$, $N⩾3$, 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}.

22 pages, 1 figure