Gradient bounds for nonlinear degenerate parabolic equations and application to large time behavior of systems

Nonlinear Analysis, Theory, Methods and Applications

Volumn 130, Issue , 2016, Pages 76-101

  Ley, Olivier ^a   Nguyen, Vinh Duc ^b  

^a UNIVERSITÉ DE RENNES 1   (France)

^b UNIVERSITÉ PARIS EST CRÉTEIL   (France)

Author keywords

Asymptotic behavior; Ergodic problem; Gradient bounds; Hamilton Jacobi equations; Monotone systems; Nonlinear degenerate elliptic equations; Nonlinear degenerate parabolic equations; Oscillation; Strong maximum principle; Viscosity solutions

Indexed keywords

ASYMPTOTIC ANALYSIS; HAMILTONIANS; MAXIMUM PRINCIPLE; PARTIAL DIFFERENTIAL EQUATIONS; VISCOSITY;

ASYMPTOTIC BEHAVIORS; DEGENERATE ELLIPTIC EQUATION; ERGODICS; GRADIENT BOUNDS; HAMILTON - JACOBI EQUATIONS; MONOTONE SYSTEMS; NONLINEAR DEGENERATE PARABOLIC EQUATIONS; OSCILLATION; STRONG MAXIMUM PRINCIPLE; VISCOSITY SOLUTIONS;

NONLINEAR EQUATIONS;

EID: 84944810383     PISSN: 0362546X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.na.2015.09.012     Document Type: Article

References (29)

1. M. Arisawa, and P.-L. Lions On ergodic stochastic control Comm. Partial Differential Equations 23 11-12 1998 2187 2217
2. G. Barles A weak Bernstein method for fully nonlinear elliptic equations Differential Integral Equations 4 2 1991 241 262
3. C0,α-regularity and estimates for solutions of elliptic and parabolic equations by the Ishii & Lions method GAKUTO Internat. Ser. Math. Sci. Appl. 30 2008 33 47
4. C0,α-regularity of viscosity subsolutions for superquadratic viscous Hamilton-Jacobi equations and applications Nonlinear Anal. 73 1 2010 31 47
5. G. Barles, H. Ishii, and H. Mitake A new PDE approach to the large time asymptotics of solutions of Hamilton-Jacobi equations Bull. Math. Sci. 3 2013 363 388
6. G. Barles, and P.E. Souganidis On the large time behavior of solutions of Hamilton-Jacobi equations SIAM J. Math. Anal. 31 4 2000 925 939 (electronic)
7. 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 6 2001 1311 1323 (electronic)
8. F. Cagnetti, D. Gomes, H. Mitake, and H.V. Tran A new method for large time behavior of convex Hamilton-Jacobi equations I: degenerate equations and weakly coupled systems Ann. Inst. H. Poincaré Anal. Non Linéaire 32 1 2015 183 200
9. F. Camilli, O. Ley, P. Loreti, and V.D. Nguyen Large time behavior of weakly coupled systems of first-order Hamilton-Jacobi equations NoDEA Nonlinear Differential Equations Appl. 19 6 2012 719 749
10. I. Capuzzo Dolcetta, F. Leoni, and A. Porretta Hölder estimates for degenerate elliptic equations with coercive Hamiltonians Trans. Amer. Math. Soc. 362 9 2010 4511 4536
11. P. Cardaliaguet A note on the regularity of solutions of Hamilton-Jacobi equations with superlinear growth in the gradient variable ESAIM Control Optim. Calc. Var. 15 2 2009 367 376
12. P. Cardaliaguet, and L. Silvestre Hölder continuity to Hamilton-Jacobi equations with superquadratic growth in the gradient and unbounded right-hand side Comm. Partial Differential Equations 37 9 2012 1668 1688
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. (NS) 27 1 1992 1 67
14. F. Da Lio Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations Commun. Pure Appl. Anal. 3 3 2004 395 415
15. A. Davini, and A. Siconolfi A generalized dynamical approach to the large time behavior of solutions of Hamilton-Jacobi equations SIAM J. Math. Anal. 38 2 2006 478 502 (electronic)
16. L.C. Evans Adjoint and compensated compactness methods for Hamilton-Jacobi PDE Arch. Ration. Mech. Anal. 197 3 2010 1053 1088
17. A. Fathi Sur la convergence du semi-groupe de Lax-Oleinik C. R. Acad. Sci. Paris Sér. I 327 3 1998 267 270
18. W.H. Fleming, and Q. Zhang Risk-sensitive production planning of a stochastic manufacturing system SIAM J. Control Optim. 36 4 1998 1147 1170 (electronic)
19. H. Ishii Perron's method for Hamilton-Jacobi equations Duke Math. J. 55 2 1987 369 384
20. H. Ishii, and S. Koike Viscosity solutions for monotone systems of second-order elliptic PDEs Comm. Partial Differential Equations 16 6-7 1991 1095 1128
21. H. Ishii, and P.-L. Lions Viscosity solutions of fully nonlinear second-order elliptic partial differential equations J. Differential Equations 83 1 1990 26 78
22. O. Ley, and V.D. Nguyen Large time behavior for nonlinear degenerate parabolic equations J. Math. Pures Appl. 102 2 2014 293 314
23. P.-L. Lions Generalized Solutions of Hamilton-Jacobi Equations 1982 Pitman (Advanced Publishing Program) Boston, Mass.
24. P.-L. Lions, B. Papanicolaou, S.R.S. Varadhan, Homogenization of Hamilton-Jacobi equations, (unpublished) 1986.
25. P.-L. Lions, and P.E. Souganidis Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications Ann. Inst. H. Poincaré Anal. Non Linéaire 22 5 2005 667 677
26. H. Mitake, and H.V. Tran Remarks on the large time behavior of viscosity solutions of quasi-monotone weakly coupled systems of Hamilton-Jacobi equations Asymptot. Anal. 77 1-2 2012 43 70
27. H. Mitake, and H.V. Tran A dynamical approach to the large-time behavior of solutions to weakly coupled systems of Hamilton-Jacobi equations J. Math. Pures Appl. 101 2014 76 93
28. G. Namah, and J.-M. Roquejoffre Remarks on the long time behaviour of the solutions of Hamilton-Jacobi equations Comm. Partial Differential Equations 24 5-6 1999 883 893
29. V.D. Nguyen Some results on the large time behavior of weakly coupled systems of first-order hamilton-jacobi equations J. Evol. Equ. 14 2014 299 331