Homogenization of a semilinear heat equation

Annalisa Cesaroni; Nicolas Dirr; Matteo Novaga

doi:10.5802/jep.54

Homogenization of a semilinear heat equation
[Homogénéisation d’une équation de la chaleur semi-linéaire]

Annalisa Cesaroni ¹ ; Nicolas Dirr ² ; Matteo Novaga ³

¹ Dipartimento di Scienze Statistiche, Università di Padova Via Cesare Battisti, 241/243, 35121 Padova, Italy
² Cardiff School of Mathematics, Cardiff University Senghennydd Road, Cardiff, CF24 4AG, UK
³ Dipartimento di Matematica, Università di Pisa Largo Bruno Pontecorvo 5, 56127 Pisa, Italy

Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 633-660.

Résumé
Abstract

Nous considérons l’homogénéisation d’une équation de la chaleur semi-linéaire avec viscosité tendant vers $0$ et un potentiel positif oscillant dépendant de $u / ε$ . Suivant le rapport entre la fréquence des oscillations dans le potentiel et le facteur tendant vers $0$ dans la viscosité, nous obtenons différents régimes de l’évolution limite et nous discutons la convergence uniforme locale des solutions du problème effectif. L’aspect intéressant du modèle est que, dans un régime à forte diffusion, l’opérateur effectif est discontinu comme fonction du gradient. Nous obtenons une caractérisation complète de la solution limite en dimension $n = 1$ , alors qu’en dimension $n > 1$ nous analysons les propriétés principales des solution du problème effectif sélectionné à la limite, et nous montrons l’unicité pour certaines classes de données initiales.

We consider the homogenization of a semilinear heat equation with vanishing viscosity and with oscillating positive potential depending on $u / ε$ . According to the rate between the frequency of oscillations in the potential and the vanishing factor in the viscosity, we obtain different regimes in the limit evolution and we discuss the locally uniform convergence of the solutions to the effective problem. The interesting feature of the model is that in the strong diffusion regime the effective operator is discontinuous in the gradient entry. We get a complete characterization of the limit solution in dimension $n = 1$ , whereas in dimension $n > 1$ we discuss the main properties of the solutions to the effective problem selected at the limit and we prove uniqueness for some classes of initial data.

Reçu le : 2016-07-08
Accepté le : 2017-05-23
Publié le : 2017-05-30

MR Zbl

DOI : 10.5802/jep.54

Classification : 35B27, 74Q10, 35D40
Keywords: Homogenization, parabolic equations, vanishing viscosity
Mots-clés : Homogénéisation, équations paraboliques, viscosité tendant vers $0$

Affiliations des auteurs :

Annalisa Cesaroni ¹ ; Nicolas Dirr ² ; Matteo Novaga ³

¹ Dipartimento di Scienze Statistiche, Università di Padova Via Cesare Battisti, 241/243, 35121 Padova, Italy
² Cardiff School of Mathematics, Cardiff University Senghennydd Road, Cardiff, CF24 4AG, UK
³ Dipartimento di Matematica, Università di Pisa Largo Bruno Pontecorvo 5, 56127 Pisa, Italy

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Homogenization of a semilinear heat equation},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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Annalisa Cesaroni; Nicolas Dirr; Matteo Novaga. Homogenization of a semilinear heat equation. Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 633-660. doi : 10.5802/jep.54. https://jep.centre-mersenne.org/articles/10.5802/jep.54/

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