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H 1 Dynamics of bounded solutions of parabolic equations on the real line - Part 1

Auteurs : Polacik, Peter (Auteur de la Conférence)
CIRM (Editeur )

Résumé : We consider parabolic equations of the form $u_t = u_{xx} + f (u)$ on the real line. Unlike their counterparts on bounded intervals, these equations admit bounded solutions whose large-time dynamics is not governed by steady states. Even with respect to the locally uniform convergence, the solutions may not be quasiconvergent, that is, their omega-limit sets may contain nonstationary solutions.
We will start this lecture series by exhibiting several examples of non-quasiconvergent solutions, discussing also some entire solutions appearing in their omega-limit sets. Minimal assumptions on the nonlinearity are needed in the examples, which shows that non-quasiconvergent solutions occur very frequently in this type of equations. Our next goal will be to identify specific classes of initial data that lead to quasiconvergent solutions. These include localized initial data (joint work with Hiroshi Matano) and front-like initial data. Finally, in the last part of these lectures, we take a more global look at the solutions with such initial data. Employing propagating terraces, or stacked families of traveling fronts, we describe their entire spatial profile at large times.

Codes MSC :
35B40 - Asymptotic behavior of solutions of PDE
35K15 - Initial value problems for second-order parabolic equations
35K55 - Nonlinear parabolic equations

 Informations sur la Vidéo Langue : Anglais Date de publication : 07/04/16 Date de captation : 22/03/16 Collection : Research talks ; Partial Differential Equations ; Dynamical Systems and Ordinary Differential Equations Format : MP4 Durée : 00:58:29 Domaine : Dynamical Systems & ODE ; PDE Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2016-03-22_Polacik.mp4 Informations sur la rencontre Nom de la rencontre : Dynamics of evolution equations / Systèmes dynamiques et problèmes d'évolutionOrganisateurs de la rencontre : Joly, Romain ; Raugel, Geneviève ; Yi, YingfeiDates : 21/03/16 - 25/03/16 Année de la rencontre : 2016 URL Congrès : http://conferences.cirm-math.fr/1335.htmlCitation Data DOI : 10.24350/CIRM.V.18949203 Cite this video as: Polacik, Peter (2016). Dynamics of bounded solutions of parabolic equations on the real line - Part 1. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.18949203 URI : http://dx.doi.org/10.24350/CIRM.V.18949203

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