JournalsaihpcVol. 38, No. 2pp. 421–450

A dynamical approach to semilinear elliptic equations

  • Christopher Jones

    Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA

  • Yuri Latushkin

    Department of Mathematics, University of Missouri, Columbia, MO 65211, USA

  • Alim Sukhtayev

    Department of Mathematics, Miami University, Oxford, OH 45056, USA

  • Margaret Beck

    Department of Mathematics and Statistics, Boston University, Boston, MA 02215, USA

  • Graham Cox

    Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NL A1C 5S7, Canada
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Abstract

A characterization of a semilinear elliptic partial differential equation (PDE) on a bounded domain in is given in terms of an infinite-dimensional dynamical system. The dynamical system is on the space of boundary data for the PDE. This is a novel approach to elliptic problems that enables the use of dynamical systems tools in studying the corresponding PDE. The dynamical system is ill-posed, meaning solutions do not exist forwards or backwards in time for generic initial data. We offer a framework in which this ill-posed system can be analyzed. This can be viewed as generalizing the theory of spatial dynamics, which applies to the case of an infinite cylindrical domain.

Cite this article

Christopher Jones, Yuri Latushkin, Alim Sukhtayev, Margaret Beck, Graham Cox, A dynamical approach to semilinear elliptic equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 38 (2021), no. 2, pp. 421–450

DOI 10.1016/J.ANIHPC.2020.08.001