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Contributions to Nonlinear Elliptic Equations and Systems pp 405–423Cite as

Birkhäuser

Spatial trajectories and convergence to traveling fronts for bistable reaction-diffusion equations

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Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 86))

Abstract

We consider the semilinear parabolic equation

$$\displaystyle\begin{array}{rcl} u_{t} = u_{xx} + f(u),\qquad x \in \mathbb{R},\ t > 0,& &{}\end{array}$$
(1)

where f is a bistable nonlinearity. It is well known that for a large class of initial data, the corresponding solutions converge to traveling fronts. We give a new proof of this classical result as well as some generalizations. Our proof uses a geometric method, which makes use of spatial trajectories \(\{(u(x,t),u_{x}(x,t)): x \in \mathbb{R}\}\) of solutions of (1).

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Acknowledgements

This work was supported in part by the NSF Grant DMS–1161923.

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Authors and Affiliations

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

    Peter Poláčik

Authors
  1. Peter Poláčik
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Correspondence to Peter Poláčik .

Editor information

Editors and Affiliations

  1. Universidade de São Paulo, Instituto de Ciências Matemáticas e de Computação, São Carlos, São Paulo, Brazil

    Alexandre Nolasco de Carvalho

  2. Dipartimento di Matematica, Università degli Studi di Milano, Lombardia, Milano, Italy

    Bernhard Ruf

  3. Universidade de São Paulo, Instituto de Ciências Matemáticas e de Computação, São Carlos, São Paulo, Brazil

    Ederson Moreira dos Santos

  4. Département de Mathématique. CP 214, Université Libre de Bruxelles, Brussels, Belgium

    Jean-Pierre Gossez

  5. Universidade de São Paulo, Instituto de Ciências Matemáticas e de Computação, São Carlos, São Paulo, Brazil

    Sergio Henrique Monari Soares

  6. Université Pierre et Marie Curie, Laboratoire Jacques-Louis Lions, Paris, Paris, France

    Thierry Cazenave

Additional information

Dedicated to Professor Djairo Guedes de Figueiredo on the occasion of his 80th birthday

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Poláčik, P. (2015). Spatial trajectories and convergence to traveling fronts for bistable reaction-diffusion equations. In: Nolasco de Carvalho, A., Ruf, B., Moreira dos Santos, E., Gossez, JP., Monari Soares, S., Cazenave, T. (eds) Contributions to Nonlinear Elliptic Equations and Systems. Progress in Nonlinear Differential Equations and Their Applications, vol 86. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-19902-3_24

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