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

Peter Poláčik

doi:10.1007/978-3-319-64173-7_11

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

Peter Poláčik

School of Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

10 Scopus citations

Abstract

We consider semilinear parabolic equations u_t= u_xx+ 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 language	English (US)
Title of host publication	Patterns of Dynamics - In Honour of Bernold Fiedler’s 60th Birthday
Editors	Pavel Gurevich, Juliette Hell, Arnd Scheel, Bjorn Sandstede
Publisher	Springer New York LLC
Pages	172-183
Number of pages	12
ISBN (Print)	9783319641720
DOIs	https://doi.org/10.1007/978-3-319-64173-7_11
State	Published - 2017
Event	Conference on Patterns of Dynamics held in honor of Bernold Fiedler’s 60th Birthday, 2016 - Berlin, Germany Duration: Jul 25 2016 → Jul 29 2016

Publication series

Name	Springer Proceedings in Mathematics and Statistics
Volume	205
ISSN (Print)	2194-1009
ISSN (Electronic)	2194-1017

Other

Other	Conference on Patterns of Dynamics held in honor of Bernold Fiedler’s 60th Birthday, 2016
Country/Territory	Germany
City	Berlin
Period	7/25/16 → 7/29/16

Bibliographical note

Publisher Copyright:
© Springer International Publishing AG 2017.

Keywords

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

Publisher link

10.1007/978-3-319-64173-7_11

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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

Convergence and Quasiconvergence Properties of Solutions of Parabolic Equations on the Real Line: An Overview. / Poláčik, Peter.
Patterns of Dynamics - In Honour of Bernold Fiedler’s 60th Birthday. ed. / Pavel Gurevich; Juliette Hell; Arnd Scheel; Bjorn Sandstede. Springer New York LLC, 2017. p. 172-183 (Springer Proceedings in Mathematics and Statistics; Vol. 205).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

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. Springer Proceedings in Mathematics and Statistics, vol. 205, Springer New York LLC, pp. 172-183, Conference on Patterns of Dynamics held in honor of Bernold Fiedler’s 60th Birthday, 2016, Berlin, Germany, 7/25/16. https://doi.org/10.1007/978-3-319-64173-7_11

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

Poláčik, Peter. / Convergence and Quasiconvergence Properties of Solutions of Parabolic Equations on the Real Line : An Overview. Patterns of Dynamics - In Honour of Bernold Fiedler’s 60th Birthday. editor / Pavel Gurevich ; Juliette Hell ; Arnd Scheel ; Bjorn Sandstede. Springer New York LLC, 2017. pp. 172-183 (Springer Proceedings in Mathematics and Statistics).

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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.",

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AB - 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.

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KW - Entire solutions

KW - Quasiconvergence

KW - Semilinear heat equation on the real line

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

Abstract

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Keywords

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