Geodesic
A dynamical approach to semilinear elliptic equations
Beck, Margaret1 ; Cox, Graham2 ; Jones, Christopher3 ; Latushkin, Yuri4 ; Sukhtayev, Alim5
1a Department of Mathematics and Statistics, Boston University, Boston, MA 02215, USA
2b Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NL A1C 5S7, Canada
3c Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA
4d Department of Mathematics, University of Missouri, Columbia, MO 65211, USA
5e 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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A characterization of a semilinear elliptic partial differential equation (PDE) on a bounded domain in Rn 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.

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DOI : 10.1016/j.anihpc.2020.08.001
Classification : 35J67, 35A24, 34D09, 35J25
Keywords: Semilinear equations, Spatial dynamics, Dynamical systems

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 
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     title = {A dynamical approach to semilinear elliptic equations},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {421--450},
     year = {2021},
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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

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