Geodesic
A dynamical approach to semilinear elliptic equations
Beck, Margaret1 ; Cox, Graham2 ; Jones, Christopher3 ; Latushkin, Yuri4 ; Sukhtayev, Alim5
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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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},
     publisher = {Elsevier},
     volume = {38},
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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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