Large time behavior of periodic viscosity solutions for uniformly parabolic integro-differential equations

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 λ t + v ( x ) . We then prove t...

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Bibliographic Details
Published in:Calculus of variations and partial differential equations 2014-05, Vol.50 (1-2), p.283-304
Main Authors: Barles, Guy, Chasseigne, Emmanuel, Ciomaga, Adina, Imbert, Cyril
Format: Article
Language:English
Subjects:
Analysis
Calculus of variations
Calculus of Variations and Optimal Control
Optimization
Cauchy problem
Control
Differential equations
Estimates
Handles
Infinity
Mathematical analysis
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Nonlinear equations
Operators
Systems Theory
Texts
Theoretical
Viscosity
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Summary: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 λ 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: (1) 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 (2) 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.
ISSN:0944-2669
1432-0835
DOI:10.1007/s00526-013-0636-2