Multi-parameter perturbations for the space-periodic heat equation

Matteo Dalla Riva; Paolo Luzzini; Riccardo Molinarolo; Paolo Musolino

doi:10.3934/cpaa.2024004

Article Contents

2024, Volume 23, Issue 2: 144-164. Doi: 10.3934/cpaa.2024004

This issue Previous Article Next Article Existence of sign-changing solutions for the Hénon type parabolic equation with singular initial data

Multi-parameter perturbations for the space-periodic heat equation

1.
Dipartimento di Ingegneria, Università degli Studi di Palermo, Viale delle Scienze, Ed. 8, 90128 Palermo, Italy
2.
Dipartimento di Matematica 'Tullio Levi-Civita', Università degli Studi di Padova, Via Trieste 63, 35121 Padova, Italy
3.
Dipartimento di Scienze Molecolari e Nanosistemi, Università Ca' Foscari Venezia, Via Torino 155, 30172 Venezia Mestre, Italy

^*Corresponding author: Paolo Musolino
^*Corresponding author: Paolo Musolino

Received: September 2023

Revised: November 2023

Early access: January 2024

Published: February 2024

M. D. R., P. L. and P. M. are supported by the "INdAM GNAMPA Project" codice CUP_E53C22001930001 "Operatori differenziali e integrali in geometria spettrale" and of the project funded by the EuropeanUnion – NextGenerationEU under the National Recovery and Resilience Plan (NRRP), Mission 4 Component 2 Investment 1.1 - Call PRIN 2022 No. 104 of February 2, 2022 of Italian Ministry of University and Research; Project 2022SENJZ3 (subject area: PE - Physical Sciences and Engineering) "Perturbation problems and asymptotics for elliptic differential equations: variational and potential theoretic methods". P. M. and R. M. are also supported by the SPIN Project "DOMain perturbation problems and INteractions Of scales - DOMINO" of the Ca' Foscari University of Venice and the support from EU through the H2020-MSCA-RISE-2020 project EffectFact, Grant agreement ID: 101008140.

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Abstract

This paper is divided into three parts. The first part focuses on periodic layer heat potentials, demonstrating their smooth dependence on regular perturbations of the support of integration. In the second part, we present an application of the results from the first part. Specifically, we consider a transmission problem for the heat equation in a periodic domain and we show that the solution depends smoothly on the shape of the transmission interface, boundary data, and transmission parameters. Finally, in the last part of the paper, we fix all parameters except for the transmission parameters and outline a strategy to deduce an explicit expansion of the solution using Neumann-type series.

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Mathematics Subject Classification: Primary: 35K20; 31B10. Secondary: 35B10; 47H30; 45A05.

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Figure 1. The sets $ \mathbb{S}[\phi]^- $, $ \mathbb{S}[\phi] $, and $ \phi(\partial\Omega) $ in case $ n = 2 $

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