Fractional oscillon equations; solvability and connection with classical oscillon equations

Flank D. M. Bezerra; Rodiak N. Figueroa-López; Marcelo J. D. Nascimento

doi:10.3934/cpaa.2021067

Article Contents

2021, Volume 20, Issue 6: 2257-2277. Doi: 10.3934/cpaa.2021067

This issue Previous Article Positive solutions for Choquard equation in exterior domains Next Article Local existence and uniqueness in Sobolev spaces for first-order conformal causal relativistic viscous hydrodynamics

Fractional oscillon equations; solvability and connection with classical oscillon equations

1.
Universidade Federal da Paraíba, Departamento de Matemática, 58051-900 João Pessoa PB, Brazil
2.
Departamento de Matemática, Universidade Federal de São Carlos, 13565-905 São Carlos SP, Brazil

^* Corresponding author
^* Corresponding author

Received: February 29, 2020

Revised: February 28, 2021

Early access: April 2021

Published: June 2021

The second author is supported by CAPES/Finance Code 001/2019, Brazil. The third author is partially supported by FAPESP grant # 2017/06582-2, Brazil

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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\geq 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 [6], and Di Plinio, Duane, Temam [10].

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Mathematics Subject Classification: Primary: 37B55, 35B40, 35B41; Secondary: 34A08, 35L71.

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Figure 1. Partial description of the fractional power spaces scale for $ \varLambda(t) , t\in{\mathbb{R}} $.

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References

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