PDE problems with concentrating terms near the boundary

Ángela Jiménez-Casas; Aníbal Rodríguez-Bernal

doi:10.3934/cpaa.2020095

Article Contents

2020, Volume 19, Issue 4: 2147-2195. Doi: 10.3934/cpaa.2020095

This issue Previous Article Pullback attractors for 2D Navier–Stokes equations with delays and the flattening property Next Article Chain recurrence and structure of

$ \omega $

-limit sets of multivalued semiflows

PDE problems with concentrating terms near the boundary

Ángela Jiménez-Casas^1, and
Aníbal Rodríguez-Bernal^2,3, ,

1.
Grupo de Dinámica No Lineal, Universidad Pontificia Comillas de Madrid, C/ Alberto Aguilera 23, 28015 Madrid, Spain
2.
Departamento de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid. 28040 Madrid, Spain
3.
Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM

^*Corresponding author
^*Corresponding author

Dedicated to Professor Tom´as Caraballo on occasion of his Sixtieth Birthday

Received: April 30, 2019

Revised: August 31, 2019

Published: January 2020

Partially supported by Project MTM2016-75465, MINECO, Spain and FIS2016-78883-C2-2-P(AEI/FEDER, U.E.). Partially supported by Severo Ochoa project SEV-2015-0554 (MINECO)

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Abstract

In this paper we study several PDE problems where certain linear or nonlinear termsin the equation concentrate in the domain, typically (but not exclusively) near the boundary. We analyze some linear and nonlinear elliptic models, linear and nonlinear parabolic ones as well as some damped wave equations. We show that in all these singularly perturbed problems, the concentrating terms give rise in the limit to a modification in the original boundary condition of the problem. Hence we describe in each case which is the singular limit problem and analyze the convergence of solutions.

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Mathematics Subject Classification: Primary: 35B25, 35B40, 35J25, 35L15, 35K20, 35P30; Secondary: 35B40, 35P99.

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