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

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Patterns of Dynamics (PaDy 2016)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 205))

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Abstract

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

Dedicated to Bernold Fiedler on the occasion of his 60th birthday.

Supported in part by the NSF Grant DMS-1565388.

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Authors and Affiliations

  1. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Peter Poláčik

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  1. Peter Poláčik
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Correspondence to Peter Poláčik .

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Editors and Affiliations

  1. Mathematical Institute, Free University of Berlin, Berlin, Germany

    Pavel Gurevich

  2. Mathematical Institute, Free University of Berlin, Berlin, Germany

    Juliette Hell

  3. Division of Applied Mathematics, Brown University, Providence, Rhode Island, USA

    Björn Sandstede

  4. School of Mathematics, University of Minnesota, Minneapolis, Minnesota, USA

    Arnd Scheel

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Poláčik, P. (2017). Convergence and Quasiconvergence Properties of Solutions of Parabolic Equations on the Real Line: An Overview. In: Gurevich, P., Hell, J., Sandstede, B., Scheel, A. (eds) Patterns of Dynamics. PaDy 2016. Springer Proceedings in Mathematics & Statistics, vol 205. Springer, Cham. https://doi.org/10.1007/978-3-319-64173-7_11

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