Abstract
Using an operator approach, we discuss stationary solutions to Fokker-Planck equations and systems with nonlinear reaction terms. The existence of solutions is obtained by using Banach, Schauder and Schaefer fixed point theorems, and for systems by means of Perov’s fixed point theorem. Using the Ekeland variational principle, it is proved that the unique solution of the problem minimizes the energy functional, and in case of a system that it is the Nash equilibrium of the energy functionals associated to the component equations.
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Acknowledgments
The authors would like to express their gratitude to the anonymous reviewers for their time and effort for reading the manuscript and valuable comments and suggestions.
This research is carried out within the national group GNAMPA of INdAM and supported by the INdAM-GNAMPA Project 2019 Metodi topologici per problemi differenziali nonlineari ed applicazioni.
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Precup, R., Rubbioni, P. Stationary Solutions of Fokker-Planck Equations with Nonlinear Reaction Terms in Bounded Domains. Potential Anal 57, 181–199 (2022). https://doi.org/10.1007/s11118-021-09911-6
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DOI: https://doi.org/10.1007/s11118-021-09911-6
Keywords
- Elliptic equation
- Reaction-diffusion equation
- Semi-linear Fokker-Planck equation
- Fixed point
- Variational method
- Nash type equilibrium