Multi-scale multi-profile global solutions of parabolic equations in$\mathbb{R}^N $

Thierry Cazenave; Flávio Dickstein; Fred B. Weissler

doi:10.3934/dcdss.2012.5.449

Article Contents

2012, Volume 5, Issue 3: 449-472. Doi: 10.3934/dcdss.2012.5.449

This issue Previous Article An existence theorem for the magneto-viscoelastic problem Next Article Korn's inequalities: The linear vs. the nonlinear case

Multi-scale multi-profile global solutions of parabolic equations in $\mathbb{R}^N $

1.
Université Pierre et Marie Curie & CNRS, Laboratoire Jacques-Louis Lions, B.C. 187, 4 place Jussieu, 75252 Paris cedex 05
2.
Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, 21944–970 Rio de Janeiro, R.J.
3.
Université Paris 13, CNRS UMR 7539 LAGA, 99 Avenue J.-B. Clément, F-93430 Villetaneuse

Received: August 31, 2010

Revised: May 31, 2011

Published: May 2012

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Abstract

This paper explores certain concepts which extend the notions of (forward) self-similar and asymptotically self-similar solutions. A self-similar solution of an evolution equation has the property of being invariant with respect to a certain group of space-time dilations. An asymptotically self-similar solution approaches (in an appropriate sense) a self-similar solution to first order approximation for large time. Such solutions have a definite long-time asymptotic behavior, with respect to a specific time dependent spatial rescaling. After reviewing these fundamental concepts and the basic known results for heat equations on $\mathbb{R}^N $, we examine the possibility that a global solution might not be asymptotically self-similar. More precisely, we show that the asymptotic form of a solution can evolve differently along different time sequences going to infinity. Indeed, there exist solutions which are asymptotic to infinitely many different self-similar solutions, along different time sequences, all with respect to the same time dependent rescaling. We exhibit an explicit relationship between this phenomenon and the spatial asymptotic behavior of the initial value under a related group of dilations. In addition, we show that a given solution can exhibit nontrivial asymptotic behavior along different time sequences going to infinity, and with respect to different time dependent rescalings.

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Mathematics Subject Classification: 35K05, 35K58, 35B40.

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