Large Time Behavior of Periodic Viscosity Solutions for Uniformly Parabolic Integro-Differential Equations

Guy Barles 1, 2 Emmanuel Chasseigne 2, 1 Adina Ciomaga 3 Cyril Imbert 4
1 LMPT - Laboratoire de Mathématiques et Physique Théorique
2 FDP - Fédération de recherche Denis Poisson
3 Department of Mathematics
4 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
Abstract : In this paper, we study the large time behavior of solutions of a class of parabolic fully nonlinear integro-differential equations in a periodic setting. In order to do so, we first solve the ergodic problem}(or cell problem), i.e. we construct solutions of the form $\lambda t + v(x)$. We then prove that solutions of the Cauchy problem look like those specific solutions as time goes to infinity. We face two key difficulties to carry out this classical program: (i) the fact that we handle the case of ''mixed operators'' for which the required ellipticity comes from a combination of the properties of the local and nonlocal terms and (ii) the treatment of the superlinear case (in the gradient variable). Lipschitz estimates previously proved by the authors (2012) and Strong Maximum principles proved by the third author (2012) play a crucial role in the analysis.
keyword : Parabolic nonlinear integro-differential equations Ergodic problem Long time behavior Strong maximum principle Lipschitz estimates
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Calc. Var. Partial Differential Equations, 2014, 50 (1-2), pp.283-304. 〈10.1007/s00526-013-0636-2〉
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Guy Barles, Emmanuel Chasseigne, Adina Ciomaga, Cyril Imbert. Large Time Behavior of Periodic Viscosity Solutions for Uniformly Parabolic Integro-Differential Equations. Calc. Var. Partial Differential Equations, 2014, 50 (1-2), pp.283-304. 〈10.1007/s00526-013-0636-2〉. 〈hal-00743751〉

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