Large time behavior of solutions of local and nonlocal nondegenerate Hamilton-Jacobi equations with Ornstein-Uhlenbeck operator

Thi Tuyen Nguyen

doi:10.3934/cpaa.2019049

Article Contents

2019, Volume 18, Issue 3: 999-1021. Doi: 10.3934/cpaa.2019049

This issue Previous Article Next Article Uniqueness for Neumann problems for nonlinear elliptic equations

Large time behavior of solutions of local and nonlocal nondegenerate Hamilton-Jacobi equations with Ornstein-Uhlenbeck operator

Thi Tuyen Nguyen^1,2,

1.
Laboratoire de Mathématiques de l'INSA de Rouen, 685 Avenue de l'Université, 76800 Saint-Étienne-du-Rouvray, France
2.
On leave from IRMAR, Université de Rennes 1, France

Received: May 31, 2017

Revised: February 28, 2018

Published: May 2019

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Abstract

We study the well-posedness of second order Hamilton-Jacobi equations with an Ornstein-Uhlenbeck operator in $\mathbb{R}^N$ and $\mathbb{R}^N× [0, +∞).$ As applications, we solve the associated ergodic problem associated to the stationary equation and obtain the large time behavior of the solutions of the evolution equation when it is nondegenerate. These results are some generalizations of the ones obtained by Fujita, Ishii & Loreti 2006 [19] by considering more general diffusion matrices or nonlocal operators of integro-differential type and general sublinear Hamiltonians. Our work uses as a key ingredient the a-priori Lipschitz estimates obtained in Chasseigne, Ley & Nguyen 2017 [10].

Keywords:

Nonlinear partial differential equations,
integro-partial differential equations,
Ornstein-Uhlenbeck operator,
Hamilton-Jacobi equations,
well-posedness,
large time behavior,
strong maximum principle.

Mathematics Subject Classification: Primary: 35B40; Secondary: 49K40, 35B50, 35D40.

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References

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