A dynamical approach to semilinear elliptic equations

Beck, Margaret; Cox, Graham; Jones, Christopher; Latushkin, Yuri; Sukhtayev, Alim

doi:10.1016/j.anihpc.2020.08.001

Beck, Margaret ¹ ; Cox, Graham ² ; Jones, Christopher ³ ; Latushkin, Yuri ⁴ ; Sukhtayev, Alim ⁵

¹ a Department of Mathematics and Statistics, Boston University, Boston, MA 02215, USA
² b Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NL A1C 5S7, Canada
³ c Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA
⁴ d Department of Mathematics, University of Missouri, Columbia, MO 65211, USA
⁵ e Department of Mathematics, Miami University, Oxford, OH 45056, USA

Annales de l'I.H.P. Analyse non linéaire, mars – avril 2021, Tome 38 (2021) no. 2, pp. 421-450.

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Résumé

A characterization of a semilinear elliptic partial differential equation (PDE) on a bounded domain in $R^{n}$ 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.

Reçu le : 2019-08-02
Révisé le : 2020-03-24
Accepté le : 2020-08-04

MR Zbl

DOI : 10.1016/j.anihpc.2020.08.001

Classification : 35J67, 35A24, 34D09, 35J25
Mots-clés : Semilinear equations, Spatial dynamics, Dynamical systems

Affiliations des auteurs :

Beck, Margaret ¹ ; Cox, Graham ² ; Jones, Christopher ³ ; Latushkin, Yuri ⁴ ; Sukhtayev, Alim ⁵

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     title = {A dynamical approach to semilinear elliptic equations},
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Beck, Margaret; Cox, Graham; Jones, Christopher; Latushkin, Yuri; Sukhtayev, Alim. A dynamical approach to semilinear elliptic equations. Annales de l'I.H.P. Analyse non linéaire, mars – avril 2021, Tome 38 (2021) no. 2, pp. 421-450. doi : 10.1016/j.anihpc.2020.08.001. http://www.numdam.org/articles/10.1016/j.anihpc.2020.08.001/

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