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Nonlinear Equations: Methods, Models and Applications pp 245–256Cite as

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Solutions of an Allen-Cahn Model Equation

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Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 54))

Abstract

The goal of this note is to describe some ongoing work of the authors on an Allen-Cahn type model equation. The equation serves as a model for phase transitions in a binary metallic alloy. Consider the partial differential equation

The functionFis a double well potential with a pair of global minima, e.g. witha(x,y)positive. The minima u = 0 and u = 1 ofFare calledpure states.Solutions of (PDE) with 0 <u(x, y) < 1 are calledmixed states.It will be assumed here thatFsatisfies

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References

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Wisconsin-Madison, Madison, WI, 53706, USA

    Paul H. Rabinowitz

  2. Department of Mathematics, University of Wisconsin-Richland, Richland Center, WI, 53581, USA

    Ed Stredulinsky

Authors
  1. Paul H. Rabinowitz
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  2. Ed Stredulinsky
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Editor information

Editors and Affiliations

  1. Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133, Milano, Italy

    Daniela Lupo  & Carlo D. Pagani  & 

  2. Dipartimento di Matematica, Università degli Studi di Milano, Via Saldini, 50, 20133, Milano, Italy

    Bernhard Ruf

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© 2003 Springer Basel AG

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Rabinowitz, P.H., Stredulinsky, E. (2003). Solutions of an Allen-Cahn Model Equation. In: Lupo, D., Pagani, C.D., Ruf, B. (eds) Nonlinear Equations: Methods, Models and Applications. Progress in Nonlinear Differential Equations and Their Applications, vol 54. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8087-9_19

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  • DOI: https://doi.org/10.1007/978-3-0348-8087-9_19

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9434-0

  • Online ISBN: 978-3-0348-8087-9

  • eBook Packages: Springer Book Archive

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