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

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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].

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.

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

References:

[1]	O. Alvarez and Agnès Tourin, Viscosity solutions of nonlinear integro-differential equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1996), 293-317. doi: 10.1016/S0294-1449(16)30106-8.
[2]	M. Bardi and Francesca Da Lio, On the strong maximum principle for fully nonlinear degenerate elliptic equations, Arch. Math. (Basel), 4 (1999), 276-285. doi: 10.1007/s000130050399.
[3]	G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi, Springer-Verlag Paris, 1994.
[4]	G. Barles, S. Biton and O. Ley, A geometrical approach to the study of unbounded solutions of quasilinear parabolic equations, Arch. Ration. Mech. Anal., 4 (2002), 287-325. doi: 10.1007/s002050200188.
[5]	G. Barles and P. E. Souganidis, Space-time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations, SIAM J. Math. Anal., 6 (2001), 1311-1323. doi: 10.1137/S0036141000369344.
[6]	G. Barles, E. Chasseigne, A. Ciomaga and C. Imbert, Lipschitz regularity of solutions for mixed integro-differential equations, J. Differential Equations, 11 (2012), 6012-6060. doi: 10.1016/j.jde.2012.02.013.
[7]	G. Barles, Emmanuel Chasseigne, Adina Ciomaga and Cyril Imbert, Large time behavior of periodic viscosity solutions for uniformly parabolic integro-differential equations, Calc. Var. Partial Differential Equations, 50 (2014), 283-304. doi: 10.1007/s00526-013-0636-2.
[8]	G. Barles and Cyril Imbert, Second-order elliptic integro-differential equations: viscosity solutions' theory revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (2008), 567-585. doi: 10.1016/j.anihpc.2007.02.007.
[9]	G. Barles, Olivier Ley and Erwin Topp, Lipschitz regularity for integro-differential equations with coercive Hamiltonians and applications to large time behavior, Nonlinearity, 30 (2017), 703-734. doi: 10.1088/1361-6544/aa527f.
[10]	E. Chasseigne, O. Ley, and T. T. Nguyen, A priori lipschitz estimates for solutions of local and nonlocal hamilton-jacobi equations with ornstein-uhlenbeck operator, Rev. Mat. Iberoam., to appear (2017).
[11]	A. Ciomaga, On the strong maximum principle for second-order nonlinear parabolic integro-differential equations, Adv. Differential Equations, 17 (2012), 635-671.
[12]	J. Coville, Remarks on the strong maximum principle for nonlocal operators, Electron. J. Differential Equations, 66 (2008), 1-10.
[13]	M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 1 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.
[14]	F. Da Lio, Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations, Commun. Pure Appl. Anal., 3 (2004), 395-415. doi: 10.3934/cpaa.2004.3.395.
[15]	F. Da Lio and O. Ley, Uniqueness Results for Second Order Bellman-Isaacs Equations under Quadratic Growth Assumptions and Applications, SIAM J. Control Optim., 45 (2006), 74-106. doi: 10.1137/S0363012904440897.
[16]	E. Di Nezza, Giampiero Palatucci and Enrico Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 5 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.
[17]	W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, New York, 1993.
[18]	Y. Fujita, H. Ishii and P. Loreti, Asymptotic solutions of Hamilton-Jacobi equations in Euclidean n space, Indiana Univ. Math. J., 5 (2006), 1671-1700. doi: 10.1512/iumj.2006.55.2813.
[19]	Y. Fujita, H. Ishii and P. Loreti, Asymptotic solutions of viscous Hamilton-Jacobi equations with Ornstein-Uhlenbeck operator, Comm. Partial Differential Equations, 31 (2006), 827-848. doi: 10.1080/03605300500358087.
[20]	Y. Fujita and P. Loreti, Long-time behavior of solutions to Hamilton-Jacobi equations with quadratic gradient term, NoDEA Nonlinear Differential Equations Appl., 6 (2009), 771-791. doi: 10.1007/s00030-009-0034-9.
[21]	N. Ichihara and S. Sheu, Large time behavior of solutions of Hamilton-Jacobi-Bellman equations with quadratic nonlinearity in gradient, Siam J. Math. Anal., 45 (2013), 279-306. doi: 10.1137/110832343.
[22]	H. Ishii, Perron's method for Hamilton-Jacobi equations, Duke Math. J., 2 (1987), 369-384. doi: 10.1215/S0012-7094-87-05521-9.
[23]	S. Koike, A Beginner's Guide to the Theory of Viscosity Solutions, Mathematical Society of Japan, Tokyo, 2004.
[24]	O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uralseva, Linear and Quasilinear Equations of Parabolic Type, American Math. Soc., Providence, RI, 1968.
[25]	O. Ley and V. D. Nguyen, Gradient bounds for nonlinear degenerate parabolic equations and application to large time behavior of systems, Nonlinear Anal., 130 (2016), 76-101. doi: 10.1016/j.na.2015.09.012.
[26]	O. Ley and V. D. Nguyen, Lipschitz regularity results for nonlinear strictly elliptic equations and applications, J. Differential Equations, 263 (2017), 4324-4354. doi: 10.1016/j.jde.2017.05.020.
[27]	P.-L. Lions, B. Papanicolaou, and S. R. S. Varadhan, Homogenization of Hamilton-Jacobi equations, Unpublished, 1986.
[28]	Thi Tuyen Nguyen, Comportement en temps long des solutions de quelques équations de Hamilton-Jacobi, du premier et second ordre, locales et non-locales, dans des cas nonp´eriodiques, Ph.D thesis, Université de Rennes 1, France, 2016.

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References:

[1]	O. Alvarez and Agnès Tourin, Viscosity solutions of nonlinear integro-differential equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1996), 293-317. doi: 10.1016/S0294-1449(16)30106-8.
[2]	M. Bardi and Francesca Da Lio, On the strong maximum principle for fully nonlinear degenerate elliptic equations, Arch. Math. (Basel), 4 (1999), 276-285. doi: 10.1007/s000130050399.
[3]	G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi, Springer-Verlag Paris, 1994.
[4]	G. Barles, S. Biton and O. Ley, A geometrical approach to the study of unbounded solutions of quasilinear parabolic equations, Arch. Ration. Mech. Anal., 4 (2002), 287-325. doi: 10.1007/s002050200188.
[5]	G. Barles and P. E. Souganidis, Space-time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations, SIAM J. Math. Anal., 6 (2001), 1311-1323. doi: 10.1137/S0036141000369344.
[6]	G. Barles, E. Chasseigne, A. Ciomaga and C. Imbert, Lipschitz regularity of solutions for mixed integro-differential equations, J. Differential Equations, 11 (2012), 6012-6060. doi: 10.1016/j.jde.2012.02.013.
[7]	G. Barles, Emmanuel Chasseigne, Adina Ciomaga and Cyril Imbert, Large time behavior of periodic viscosity solutions for uniformly parabolic integro-differential equations, Calc. Var. Partial Differential Equations, 50 (2014), 283-304. doi: 10.1007/s00526-013-0636-2.
[8]	G. Barles and Cyril Imbert, Second-order elliptic integro-differential equations: viscosity solutions' theory revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (2008), 567-585. doi: 10.1016/j.anihpc.2007.02.007.
[9]	G. Barles, Olivier Ley and Erwin Topp, Lipschitz regularity for integro-differential equations with coercive Hamiltonians and applications to large time behavior, Nonlinearity, 30 (2017), 703-734. doi: 10.1088/1361-6544/aa527f.
[10]	E. Chasseigne, O. Ley, and T. T. Nguyen, A priori lipschitz estimates for solutions of local and nonlocal hamilton-jacobi equations with ornstein-uhlenbeck operator, Rev. Mat. Iberoam., to appear (2017).
[11]	A. Ciomaga, On the strong maximum principle for second-order nonlinear parabolic integro-differential equations, Adv. Differential Equations, 17 (2012), 635-671.
[12]	J. Coville, Remarks on the strong maximum principle for nonlocal operators, Electron. J. Differential Equations, 66 (2008), 1-10.
[13]	M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 1 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.
[14]	F. Da Lio, Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations, Commun. Pure Appl. Anal., 3 (2004), 395-415. doi: 10.3934/cpaa.2004.3.395.
[15]	F. Da Lio and O. Ley, Uniqueness Results for Second Order Bellman-Isaacs Equations under Quadratic Growth Assumptions and Applications, SIAM J. Control Optim., 45 (2006), 74-106. doi: 10.1137/S0363012904440897.
[16]	E. Di Nezza, Giampiero Palatucci and Enrico Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 5 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.
[17]	W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, New York, 1993.
[18]	Y. Fujita, H. Ishii and P. Loreti, Asymptotic solutions of Hamilton-Jacobi equations in Euclidean n space, Indiana Univ. Math. J., 5 (2006), 1671-1700. doi: 10.1512/iumj.2006.55.2813.
[19]	Y. Fujita, H. Ishii and P. Loreti, Asymptotic solutions of viscous Hamilton-Jacobi equations with Ornstein-Uhlenbeck operator, Comm. Partial Differential Equations, 31 (2006), 827-848. doi: 10.1080/03605300500358087.
[20]	Y. Fujita and P. Loreti, Long-time behavior of solutions to Hamilton-Jacobi equations with quadratic gradient term, NoDEA Nonlinear Differential Equations Appl., 6 (2009), 771-791. doi: 10.1007/s00030-009-0034-9.
[21]	N. Ichihara and S. Sheu, Large time behavior of solutions of Hamilton-Jacobi-Bellman equations with quadratic nonlinearity in gradient, Siam J. Math. Anal., 45 (2013), 279-306. doi: 10.1137/110832343.
[22]	H. Ishii, Perron's method for Hamilton-Jacobi equations, Duke Math. J., 2 (1987), 369-384. doi: 10.1215/S0012-7094-87-05521-9.
[23]	S. Koike, A Beginner's Guide to the Theory of Viscosity Solutions, Mathematical Society of Japan, Tokyo, 2004.
[24]	O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uralseva, Linear and Quasilinear Equations of Parabolic Type, American Math. Soc., Providence, RI, 1968.
[25]	O. Ley and V. D. Nguyen, Gradient bounds for nonlinear degenerate parabolic equations and application to large time behavior of systems, Nonlinear Anal., 130 (2016), 76-101. doi: 10.1016/j.na.2015.09.012.
[26]	O. Ley and V. D. Nguyen, Lipschitz regularity results for nonlinear strictly elliptic equations and applications, J. Differential Equations, 263 (2017), 4324-4354. doi: 10.1016/j.jde.2017.05.020.
[27]	P.-L. Lions, B. Papanicolaou, and S. R. S. Varadhan, Homogenization of Hamilton-Jacobi equations, Unpublished, 1986.
[28]	Thi Tuyen Nguyen, Comportement en temps long des solutions de quelques équations de Hamilton-Jacobi, du premier et second ordre, locales et non-locales, dans des cas nonp´eriodiques, Ph.D thesis, Université de Rennes 1, France, 2016.

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