Elliptic and parabolic problems in thin domains with doubly weak oscillatory boundary

José M. Arrieta; Manuel Villanueva-Pesqueira

doi:10.3934/cpaa.2020083

Article Contents

2020, Volume 19, Issue 4: 1891-1914. Doi: 10.3934/cpaa.2020083

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Elliptic and parabolic problems in thin domains with doubly weak oscillatory boundary

José M. Arrieta^1,2, , and
Manuel Villanueva-Pesqueira^3,

1.
Departamento de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid
2.
Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, C/Nicolás Cabrera 13-15, Cantoblanco, 28049 Madrid, Spain
3.
Grupo de Investigación Dinámica No Lineal, ICAI, Universidad Pontificia de Comillas, Madrid, Spain

^* Corresponding author
^* Corresponding author

Dedicated to Tomás Caraballo on the occasion of his 60-th birthday

Received: July 2019

Revised: October 2019

Published: January 2020

This first author (JMA) has been partially supported by grants MTM2016-75465-P, ICMAT Severo Ochoa project SEV-2015-0554, MICINN, Spain and Grupo de Investigación CADEDIF, UCM. The second author (MVP) has been partially supported by grant MTM2016-75465-P MICINN, Spain and Grupo de Investigación CADEDIF, UCM.

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Abstract

In this work we consider higher dimensional thin domains with the property that both boundaries, bottom and top, present oscillations of weak type. We consider the Laplace operator with Neumann boundary conditions and analyze the behavior of the solutions as the thin domain shrinks to a fixed domain $ \omega\subset \mathbb{R}^n $. We obtain the convergence of the resolvent of the elliptic operators in the sense of compact convergence of operators, which in particular implies the convergence of the spectra. This convergence of the resolvent operators will allow us to conclude the global dynamics, in terms of the global attractors of a reaction diffusion equation in the thin domains. In particular, we show the upper semicontinuity of the attractors and stationary states. An important case treated is the case of a quasiperiodic situation, where the bottom and top oscillations are periodic but with period rationally independent.

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Mathematics Subject Classification: 35B27, 35B40, 35B41, 74K10, 74Q05.

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Figure 1. Thin domain $ R^ \epsilon $ with doubly weak oscillatory boundary

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Figure 2. Thin domain $ R_a^\epsilon $ obtained from $ R^ \epsilon $ of Figure 1

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