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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 |
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 [
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. |
show all references
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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