# American Institute of Mathematical Sciences

May  2019, 18(3): 999-1021. doi: 10.3934/cpaa.2019049

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

 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  June 2017 Revised  March 2018 Published  November 2018

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].

Citation: Thi Tuyen Nguyen. Large time behavior of solutions of local and nonlocal nondegenerate Hamilton-Jacobi equations with Ornstein-Uhlenbeck operator. Communications on Pure & Applied Analysis, 2019, 18 (3) : 999-1021. doi: 10.3934/cpaa.2019049
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