https://cnrs.hal.science/hal-03721565Hamel, FrançoisFrançoisHamelI2M - Institut de Mathématiques de Marseille - AMU - Aix Marseille Université - ECM - École Centrale de Marseille - CNRS - Centre National de la Recherche ScientifiqueRossi, LucaLucaRossiSapienza University of Rome - Dipartimento di Matematica "Guido Castelnuovo" [Roma I] - UNIROMA - Università degli Studi di Roma "La Sapienza" = Sapienza University [Rome]CAMS - Centre d'Analyse et de Mathématique sociales - EHESS - École des hautes études en sciences sociales - CNRS - Centre National de la Recherche ScientifiqueSpreading sets and one-dimensional symmetry for reaction-diffusion equationsHAL CCSD2022[MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP]Hamel, FrancoisAPPEL À PROJETS GÉNÉRIQUE 2018 - Rescousse Evolutive: effets Stochastiques et Interactions avec le STress Environnemental - - RESISTE2018 - ANR-18-CE45-0019 - AAPG2018 - VALID - - Centers of Hosting and International Mathematical Encounters - - CARMIN2010 - ANR-10-LABX-0059 - LABX - VALID - 2022-07-12 17:12:092023-07-01 05:04:592022-07-13 09:05:04enSpecial issuehttps://cnrs.hal.science/hal-03721565/documentapplication/pdf1We consider reaction-diffusion equations ∂ t u = ∆u + f (u) in the whole space R N and we are interested in the large-time dynamics of solutions ranging in the interval [0, 1], with general unbounded initial support. Under the hypothesis of the existence of a traveling front connecting 0 and 1 with a positive speed, we discuss the existence of spreading speeds and spreading sets, which describe the large-time global shape of the level sets of the solutions. The spreading speed in any direction is expressed as a Freidlin-Gärtner type formula. This formula holds under general assumptions on the reaction and for solutions emanating from initial conditions with general unbounded support, whereas most of earlier results were concerned with more specific reactions and compactly supported or almost-planar initial conditions. We then investigate the local properties of the level sets at large time. Some flattening properties of the level sets of the solutions, if initially supported on subgraphs, will be presented. We also investigate the special case of asymptotically conical-shaped initial conditions. For Fisher-KPP equations, we state some asymptotic local one-dimensional and monotonicity symmetry properties for the elements of the Ω-limit set of the solutions, in the spirit of a conjecture of De Giorgi for stationary solutions of Allen-Cahn equations. Lastly, we present some logarithmic-in-time estimates of the lag of the position of the solutions with respect to that of a planar front with minimal speed, for initial conditions which are supported on subgraphs with logarithmic growth at infinity. Some related conjectures and open problems are also listed.