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July  2016, 15(4): 1251-1263. doi: 10.3934/cpaa.2016.15.1251

## On a parabolic Hamilton-Jacobi-Bellman equation degenerating at the boundary

 1 Dipartimento di Matematica, Università di Padova, Via Trieste 63, 35121 Padova, Italy 2 Dipartimento di Scienze Statistiche, Università di Padova, Via Cesare Battisti 141, 35121 Padova, Italy 3 Università degli Studi di Padova, Dipartimento di Matematica Pura ed Applicata, Via Trieste, 63 - 35121 Padova

Received  September 2015 Revised  January 2016 Published  April 2016

We derive the long time asymptotic of solutions to an evolutive Hamilton-Jacobi-Bellman equation in a bounded smooth domain, in connection with ergodic problems recently studied in [1]. Our main assumption is an appropriate degeneracy condition on the operator at the boundary. This condition is related to the characteristic boundary points for linear operators as well as to the irrelevant points for the generalized Dirichlet problem, and implies in particular that no boundary datum has to be imposed. We prove that there exists a constant $c$ such that the solutions of the evolutive problem converge uniformly, in the reference frame moving with constant velocity $c$, to a unique steady state solving a suitable ergodic problem.
Citation: Daniele Castorina, Annalisa Cesaroni, Luca Rossi. On a parabolic Hamilton-Jacobi-Bellman equation degenerating at the boundary. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1251-1263. doi: 10.3934/cpaa.2016.15.1251
##### References:
 [1] M. Bardi, A. Cesaroni and L. Rossi, Nonexistence of nonconstant solutions of some degenerate Bellman equations and applications to stochastic control, ESAIM Control Optim. Calc. Var., to appear. doi: http://dx.doi.org/10.1051/cocv/2015033. [2] G. Barles and J. Burdeau, The Dirichlet problem for semilinear second-order degenerate elliptic equations and applications to stochastic exit control problems, Comm. Part. Diff. Eq., 20 (1995), 129-178. doi: 10.1080/03605309508821090. [3] G. Barles, A. Porretta and T.T. Tchamba, On the large time behavior of solutions of the Dirichlet problem for subquadratic viscous Hamilton-Jacobi equations, J. Math. Pures Appl., 94 (2010), 497-519. doi: 10.1016/j.matpur.2010.03.006. [4] 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., 32 (2001), 1311-1323. doi: 10.1137/S0036141000369344. [5] H. Berestycki, I. Capuzzo Dolcetta, A. Porretta and L. Rossi, Maximum Principle and generalized principal eigenvalue for degenerate elliptic operators, J. Math. Pures Appl., 103 (2015), 1276-1293. doi: 10.1016/j.matpur.2014.10.012. [6] F. Cagnetti, D. Gomes, H. Mitake and H.V. Tran, A new method for large time behavior of degenerate viscous Hamilton-Jacobi equations with convex Hamiltonians, Ann. Inst. H. Poincare Anal. Non Lineaire, 32 (2015), 183-200. doi: 10.1016/j.anihpc.2013.10.005. [7] 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.), 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5. [8] F. Da Lio, Large time behavior of solutions to parabolic equations with Neumann boundary conditions, J. Math. Anal. Appl., 339 (2008), 384-398. doi: 10.1016/j.jmaa.2007.06.052. [9] G. Fichera, Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine, Atti Accad. Naz. Lincei. Mem. Cl. Sci. Fis. Mat. Nat. Sez. I., 5 (1956), 1-30. [10] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1988. [11] H. Ishii and P. Loreti, A class of stochastic optimal control problems with state constraints, Indiana Univ. Math. J., 51 (2002), 1167-1196. doi: 10.1512/iumj.2002.51.2079. [12] J.M. Lasry and P.-L. Lions, Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints. I. The model problem, Math. Ann., 283 (1989), 583-630. doi: 10.1007/BF01442856. [13] O. Ley and V.D. Nguyen, Large time behavior for some nonlinear degenerate parabolic equations, J. Math. Pures Appl., 102 (2014), 293-314. doi: 10.1016/j.matpur.2013.11.010. [14] T. Leonori and A. Porretta, Gradient bounds for elliptic problems singular at the boundary, Arch. Ration. Mech. Anal., 202 (2011), 663-705. doi: 10.1007/s00205-011-0436-9. [15] G.M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ 1996. doi: 10.1142/3302. [16] M.V. Safonov, On the classical solution of Bellman's elliptic equation, Sov. Math. Dokl., 30 (1984), 482-485. [17] N.S. Trudinger, On regularity and existence of viscosity solutions of nonlinear second order, elliptic equations, Partial differential equations and the calculus of variations, Vol. II, 939-957, Progr. Nonlinear Differential Equations Appl., 2, Birkhauser Boston, Boston, MA, 1989. [18] G. Tian and X.J. Wang, A priori estimates for fully nonlinear parabolic equations, Int. Math. Res. Notes, 169 (2012), 1-21.

show all references

##### References:
 [1] M. Bardi, A. Cesaroni and L. Rossi, Nonexistence of nonconstant solutions of some degenerate Bellman equations and applications to stochastic control, ESAIM Control Optim. Calc. Var., to appear. doi: http://dx.doi.org/10.1051/cocv/2015033. [2] G. Barles and J. Burdeau, The Dirichlet problem for semilinear second-order degenerate elliptic equations and applications to stochastic exit control problems, Comm. Part. Diff. Eq., 20 (1995), 129-178. doi: 10.1080/03605309508821090. [3] G. Barles, A. Porretta and T.T. Tchamba, On the large time behavior of solutions of the Dirichlet problem for subquadratic viscous Hamilton-Jacobi equations, J. Math. Pures Appl., 94 (2010), 497-519. doi: 10.1016/j.matpur.2010.03.006. [4] 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., 32 (2001), 1311-1323. doi: 10.1137/S0036141000369344. [5] H. Berestycki, I. Capuzzo Dolcetta, A. Porretta and L. Rossi, Maximum Principle and generalized principal eigenvalue for degenerate elliptic operators, J. Math. Pures Appl., 103 (2015), 1276-1293. doi: 10.1016/j.matpur.2014.10.012. [6] F. Cagnetti, D. Gomes, H. Mitake and H.V. Tran, A new method for large time behavior of degenerate viscous Hamilton-Jacobi equations with convex Hamiltonians, Ann. Inst. H. Poincare Anal. Non Lineaire, 32 (2015), 183-200. doi: 10.1016/j.anihpc.2013.10.005. [7] 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.), 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5. [8] F. Da Lio, Large time behavior of solutions to parabolic equations with Neumann boundary conditions, J. Math. Anal. Appl., 339 (2008), 384-398. doi: 10.1016/j.jmaa.2007.06.052. [9] G. Fichera, Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine, Atti Accad. Naz. Lincei. Mem. Cl. Sci. Fis. Mat. Nat. Sez. I., 5 (1956), 1-30. [10] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1988. [11] H. Ishii and P. Loreti, A class of stochastic optimal control problems with state constraints, Indiana Univ. Math. J., 51 (2002), 1167-1196. doi: 10.1512/iumj.2002.51.2079. [12] J.M. Lasry and P.-L. Lions, Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints. I. The model problem, Math. Ann., 283 (1989), 583-630. doi: 10.1007/BF01442856. [13] O. Ley and V.D. Nguyen, Large time behavior for some nonlinear degenerate parabolic equations, J. Math. Pures Appl., 102 (2014), 293-314. doi: 10.1016/j.matpur.2013.11.010. [14] T. Leonori and A. Porretta, Gradient bounds for elliptic problems singular at the boundary, Arch. Ration. Mech. Anal., 202 (2011), 663-705. doi: 10.1007/s00205-011-0436-9. [15] G.M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ 1996. doi: 10.1142/3302. [16] M.V. Safonov, On the classical solution of Bellman's elliptic equation, Sov. Math. Dokl., 30 (1984), 482-485. [17] N.S. Trudinger, On regularity and existence of viscosity solutions of nonlinear second order, elliptic equations, Partial differential equations and the calculus of variations, Vol. II, 939-957, Progr. Nonlinear Differential Equations Appl., 2, Birkhauser Boston, Boston, MA, 1989. [18] G. Tian and X.J. Wang, A priori estimates for fully nonlinear parabolic equations, Int. Math. Res. Notes, 169 (2012), 1-21.
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