%0 Journal Article
%T Sharp thresholds between finite spread and uniform convergence for a reaction–diffusion equation with oscillating initial data
%+ Institut Élie Cartan de Lorraine (IECL)
%+ Institut de Mathématiques de Marseille (I2M)
%A Giletti, Thomas
%A Hamel, François
%< avec comité de lecture
%@ 0022-0396
%J Journal of Differential Equations
%I Elsevier
%V 262
%N 3
%P 1461 - 1498
%8 2017-02-05
%D 2017
%Z 1510.06556
%R 10.1016/j.jde.2016.10.014
%K oscillating data
%K spreading speeds
%K reaction-diffusion equations
%K Cauchy problem
%Z 35K91 (35B40 35C07 35K15 35K57)
%Z Mathematics [math]/Analysis of PDEs [math.AP]Journal articles
%X We investigate the large-time dynamics of solutions of multi-dimensional reaction-diffusion equations with ignition type nonlinearities. We consider solutions which are in some sense locally persistent at large time and initial data which asymptotically oscillate around the ignition threshold. We show that, as time goes to infinity, any solution either converges uniformly in space to a constant state, or spreads with a finite speed uniformly in all directions. Furthermore, the transition between these two behaviors is sharp with respect to the period vector of the asymptotic profile of the initial data. We also show the convergence to planar fronts when the initial data are asymptotically periodic in one direction.
%G English
%2 https://hal.science/hal-01218913/document
%2 https://hal.science/hal-01218913/file/giletti-hamel.pdf
%L hal-01218913
%U https://hal.science/hal-01218913
%~ CNRS
%~ UNIV-AMU
%~ IECN
%~ EC-MARSEILLE
%~ OPENAIRE
%~ I2M
%~ I2M-2014-
%~ UNIV-LORRAINE
%~ TDS-MACS
%~ IECLEDP
%~ AMIDEX
%~ ANR