Freidlin–Gärtner formula and asymptotic profile in reaction-diffusion equations
[Formule de Freidlin–Gärtner et profil asymptotique dans les équations de réaction-diffusion]
Luca Rossi1
1Dipartimento di Matematica, Sapienza Università di Roma, Piazzale Aldo Moro 5, 00185 Roma, Italy
Journées équations aux dérivées partielles (2025), Exposé no. 8, 12 p.

We address the question of the large-time behavior of solutions to reaction-diffusion equations in periodic media. We start with the description of the asymptotic shape of the invasion set, which is characterized by the Freidlin–Gärtner formula. We outline a proof of the formula that holds true for general types of reaction terms. We then present some recent results, obtained in collaboration with H. Guo and F. Hamel, for (weakly) bistable equations. They include a regular version of the Freidlin–Gärtner formula and the convergence in profile towards pulsating traveling fronts for solutions with either bounded or unbounded initial support.

Nous abordons la question du comportement à long terme des solutions d’équations de réaction-diffusion dans des milieux périodiques. Nous commençons par décrire la forme asymptotique de l’ensemble d’invasion, caractérisée par la formule de Freidlin–Gärtner. Nous exposons une preuve de cette formule, qui est valable pour des termes de réaction de type général. Nous présentons ensuite des résultats récents, obtenus en collaboration avec H. Guo et F. Hamel, pour les équations (faiblement) bistables. Ils comprennent une version régulière de la formule de Freidlin–Gärtner et la convergence en profil vers les fronts de propagation pulsatoires pour les solutions ayant un support initial borné ou non borné.

Publié le :
DOI : 10.5802/jedp.699
Classification : 35B30, 35B40, 35C07, 35K57
Keywords: Reaction-diffusion equations, pulsating traveling fronts, large-time dynamics, Freidlin–Gärtner formula
Mots-clés : Équations de réaction-diffusion, fronts progressifs pulsatoires, dynamique en temps grand, formule de Freidlin–Gärtner

Luca Rossi  1

1 Dipartimento di Matematica, Sapienza Università di Roma, Piazzale Aldo Moro 5, 00185 Roma, Italy 
Luca Rossi. Freidlin–Gärtner formula and asymptotic profile in reaction-diffusion equations. Journées équations aux dérivées partielles (2025), Exposé no. 8, 12 p.. doi: 10.5802/jedp.699
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