Long-time asymptotic solutions of convex Hamilton-Jacobi equations with Neumann type boundary conditions

Article

Abstract

We study the long-time asymptotic behavior of solutions u of the Hamilton-Jacobi equation ut(x, t) + H(x, Du(x, t)) = 0 in Ω × (0, ∞), where Ω is a bounded open subset of \({\mathbb{R}^n}\), with Hamiltonian H = H(x, p) being convex and coercive in p, and establish the uniform convergence of u to an asymptotic solution as t → ∞.

Mathematics Subject Classification (2000)

35B40 35F31 35D40 37J50 49L25 

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References

  1. 1.
    Bardi, M., Capuzzo-Dolcetta, I.: Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. In: With Appendices by Maurizio Falcone and Pierpaolo Soravia. Birkhäuser Inc., Boston, MA (1997)Google Scholar
  2. 2.
    Barles G.: Asymptotic behavior of viscosity solutions of first Hamilton-Jacobi equations. Ric. Mat. 34(2), 227–260 (1985)MathSciNetMATHGoogle Scholar
  3. 3.
    Barles, G.: Solutions de viscosité des équations de Hamilton-Jacobi. In: Mathématiques and Applications. Springer, Paris (1994)Google Scholar
  4. 4.
    Barles G., Roquejoffre J.-M.: Ergodic type problems and large time behaviour of unbounded solutions of Hamilton-Jacobi equations. Comm. Partial Differ. Equ. 31(7-9), 1209–1225 (2006)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Brales G., Souganidis P.E.: On the large time behavior of solutions of Hamilton-Jacobi equations. SIAM J. Math. Anal. 31(4), 925–939 (2000)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Crandall M.G., Ishii H., Lions P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27(1), 1–67 (1992)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Davini A., Siconolfi A.: A generalized dynamical approach to the large time behavior of solutions of Hamilton-Jacobi equations. SIAM J. Math. Anal. 38(2), 478–502 (2006)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Fathi A.: Sur la convergence du semi-groupe de Lax-Oleinik. C. R. Acad. Sci. Paris Sér. I Math. 327(3), 267–270 (1998)MathSciNetMATHGoogle Scholar
  9. 9.
    Fujita Y., Ishii H., Loreti P.: Asymptotic solutions of Hamilton-Jacobi equations in Euclidean n space. Indiana Univ. Math. J. 55(5), 1671–1700 (2006)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Ichihara N., Ishii H.: Asymptotic solutions of Hamilton-Jacobi equations with semi-periodic Hamiltonians. Commun. Partial Differ. Equ. 33(4-6), 784–807 (2008)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Ichihara N., Ishii H.: The large-time behavior of solutions of Hamilton-Jacobi equations on the real line. Methods Appl. Anal. 15(2), 223–242 (2008)MathSciNetMATHGoogle Scholar
  12. 12.
    Ichihara N., Ishii H.: Long-time behavior of solutions of Hamilton–Jacobi equations with convex and coercive Hamiltonians. Arch. Ration. Mech. Anal. 194(2), 383–419 (2009)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Ishii H.: Asymptotic solutions for large time of Hamilton-Jacobi equations in Euclidean n space. Ann. Inst. H. Poincaré Anal. Non Linéaire 25(2), 231–266 (2008)MATHCrossRefGoogle Scholar
  14. 14.
    Ishii, H.: Weak KAM aspects of convex Hamilton-Jacobi equations with Neumann type boundary conditions. Preprint (2010)Google Scholar
  15. 15.
    Kruzkov S.N.: Generalized solutions of nonlinear equations of the first order with several independent variables. II (in Russian). Mat. Sb. (N.S.) 72(114), 108–134 (1967)MathSciNetGoogle Scholar
  16. 16.
    Lions, P.-L.: Generalized solutions of Hamilton-Jacobi equations. In: Research Notes in Mathematics, vol. 69. Pitman, London (1982)Google Scholar
  17. 17.
    Lions P.-L.: Neumann type boundary conditions for Hamilton-Jacobi equations. Duke Math. J. 52(4), 793–820 (1985)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Mitake H.: Asymptotic solutions of Hamilton-Jacobi equations with state constraints. Appl. Math. Optim. 58(3), 393–410 (2008)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Mitake H.: The large-time behavior of solutions of the Cauchy-Dirichlet problem for hamilton-jacobi equations. NoDEA Nonlinear Differ. Equ. Appl. 15(3), 347–362 (2008)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Namah G., Roquejoffre J.-M.: Remarks on the long time behaviour of the solutions of Hamilton-Jacobi equations. Commun. Partial Differ. Equ. 24(5-6), 883–893 (1997)MathSciNetGoogle Scholar
  21. 21.
    Roquejoffre J.-M.: Convergence to steady states or periodic solutions in a class of Hamilton-Jacobi equations. J. Math. Pures Appl. (9) 80(1), 85–104 (2001)MathSciNetMATHCrossRefGoogle Scholar

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© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Education and Integrated Arts and SciencesWaseda UniversityTokyoJapan

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