Convergence and Quasiconvergence Properties of Solutions of Parabolic Equations on the Real Line: An Overview

Research output: Chapter in Book/Report/Conference proceedingConference contribution

4 Scopus citations

Abstract

We consider semilinear parabolic equations ut= uxx+ f(u) on R. We give an overview of results on the large time behavior of bounded solutions, focusing in particular on their limit profiles as t→ ∞ with respect to the locally uniform convergence. The collection of such limit profiles, or, the ω -limit set of the solution, always contains a steady state. Questions of interest then are whether—or under what conditions—the ω -limit set consists of steady states, or even a single steady state. We give several theorems and examples pertinent to these questions.

Original languageEnglish (US)
Title of host publicationPatterns of Dynamics - In Honour of Bernold Fiedler’s 60th Birthday
EditorsPavel Gurevich, Juliette Hell, Arnd Scheel, Bjorn Sandstede
PublisherSpringer New York LLC
Pages172-183
Number of pages12
ISBN (Print)9783319641720
DOIs
StatePublished - Jan 1 2017
EventConference on Patterns of Dynamics held in honor of Bernold Fiedler’s 60th Birthday, 2016 - Berlin, Germany
Duration: Jul 25 2016Jul 29 2016

Publication series

NameSpringer Proceedings in Mathematics and Statistics
Volume205
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017

Other

OtherConference on Patterns of Dynamics held in honor of Bernold Fiedler’s 60th Birthday, 2016
CountryGermany
CityBerlin
Period7/25/167/29/16

Keywords

  • Asymptotic behavior
  • Convergence
  • Entire solutions
  • Quasiconvergence
  • Semilinear heat equation on the real line

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    Poláčik, P. (2017). Convergence and Quasiconvergence Properties of Solutions of Parabolic Equations on the Real Line: An Overview. In P. Gurevich, J. Hell, A. Scheel, & B. Sandstede (Eds.), Patterns of Dynamics - In Honour of Bernold Fiedler’s 60th Birthday (pp. 172-183). (Springer Proceedings in Mathematics and Statistics; Vol. 205). Springer New York LLC. https://doi.org/10.1007/978-3-319-64173-7_11