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Large time behavior of solutions for a class of time-dependent Hamilton-Jacobi equations

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Abstract

We study the long-time behavior of viscosity solutions for time-dependent Hamilton-Jacobi equations by the dynamical approach based on weak KAM (Kolmogorov-Arnold-Moser) theory due to Fathi. We establish a general convergence result for viscosity solutions and adherence of the graph as t→∞.

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References

  1. Arnaud M C. Convergence of the semi-group of Lax-Oleinik: A geometric point of view. Nonlinearity, 2005, 18: 1835–1840

    Article  MathSciNet  MATH  Google Scholar 

  2. Barles G. Regularity results for first order Hamilton-Jacobi equations. Differential Integral Equations, 1990, 3: 103–125

    MathSciNet  MATH  Google Scholar 

  3. Barles G, Roquejoffre J M. Ergodic type problems and large time behaviour of unbounded solutions of Hamilton-Jacobi equations. Comm Partial Differential Equations, 2006, 31: 1209–1225

    Article  MathSciNet  MATH  Google Scholar 

  4. Barles G, Souganidis P E. On the large time behavior of solutions of Hamilton-Jacobi equations. SIAM J Math Anal, 2000, 31: 925–939

    Article  MathSciNet  MATH  Google Scholar 

  5. Barles G, Souganidis P E. Space-time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations. SIAM J Math Anal, 2001, 32: 1311–1323

    Article  MathSciNet  MATH  Google Scholar 

  6. Bernard P, Roquejoffre J M. Convergence to time-periodic solutions in time-periodic Hamilton-Jacobi equations on the circle. Comm Partial Differential Equations, 2004, 29: 457–469

    Article  MathSciNet  MATH  Google Scholar 

  7. Davini A, Siconolfi A. A generalized dynamical approach to the large time behavior of solutions of Hamilton-Jacobi equations. SIAM J Math Anal, 2006, 38: 478–502

    Article  MathSciNet  MATH  Google Scholar 

  8. Fathi A. Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens. C R Math Acad Sci Paris, 1997, 324: 1043–1046

    Article  MathSciNet  Google Scholar 

  9. Fathi A. Sur la convergence du semi-groupe de Lax-Oleinik. C R Math Acad Sci Paris, 1998, 327: 267–270

    Article  MathSciNet  Google Scholar 

  10. Fathi A. The Weak KAM Theorem in Lagrangian Dynamics. Cambridge: Cambridge University Press, 2008

    Google Scholar 

  11. Fathi A, Maderna E. Weak KAM theorem on non compact manifolds. Nonlinear Differential Equations Appl, 2007, 14: 1–27

    Article  MathSciNet  MATH  Google Scholar 

  12. Fathi A, Mather J. Failure of convergence of the Lax-Oleinik semi-group in the time-periodic case. Bull Soc Math France, 2000, 128: 473–483

    MathSciNet  MATH  Google Scholar 

  13. Fathi A, Siconolfi A. PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians. Calc Var, 2005, 22: 185–228

    Article  MathSciNet  MATH  Google Scholar 

  14. Fleming W H. The cauchy problem for a nonlinear first order parial differential equation. J Differential Equation, 1969, 5: 515–530

    Article  MathSciNet  MATH  Google Scholar 

  15. Fujita Y, Ishii H, Loreti P. Asymptotic solutions of Hamilton-Jacobi equations in Euclidean n space. Indiana Univ Math J, 2006, 55: 1671–1700

    Article  MathSciNet  MATH  Google Scholar 

  16. Ichihara N, Ishii H. Asymptotic solutions of Hamilton-Jacobi equations with semi-periodic Hamiltonians. Comm Partial Differential Equations, 2008, 33: 784–807

    Article  MathSciNet  MATH  Google Scholar 

  17. Ichihara N, Ishii H. Long-time behavior of solutions of Hamilton-Jacobi equations with convex and coercive Hamiltonians. Arch Rational Mech Anal, 2009, 194: 383–419

    Article  MathSciNet  MATH  Google Scholar 

  18. Ishii H. Asymptotic solutions for large time of Hamilton-Jacobi equations in Euclidean ℝn space. Ann Inst H Poincaré Anal Non Linéaire, 2008, 25: 231–266

    Article  MathSciNet  MATH  Google Scholar 

  19. Ishii H. Long-time asymptotic solutions of convex Hamilton-Jacobi equations with Neumann type boundary conditions. Calc Var Partial Differential Equations, 2011, 42: 189–209

    Article  MathSciNet  MATH  Google Scholar 

  20. Lions P L. Generalized Solutions of Hamilton-Jacobi Equations. London: Pitman Publishing, 1982

    MATH  Google Scholar 

  21. Mather J. Action minimizing invariant measures for positive definite Lagrangian systems. Math Z, 1991, 207: 169–207

    Article  MathSciNet  MATH  Google Scholar 

  22. Namah G, Roquejoffre J M. Remarks on the long-time behaviour of the solutions of Hamilton-Jacobi equations. Comm Partial Differential Equations, 1999, 24: 883–893

    Article  MathSciNet  MATH  Google Scholar 

  23. Roquejoffre J M. Convergence to steady states or periodic solutions in a class of Hamilton-Jacobi equations. J Math Pures Appl, 2001, 80: 85–104

    Article  MathSciNet  MATH  Google Scholar 

  24. Wang K Z, Yan J. A new kind of Lax-Oleinik type operator with parameters for time-periodic positive definite Lagrangian systems. Comm Math Phys, 2012, 309: 663–691

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    QiHuai Liu & Jun Yan

  2. School of Mathematics and Computing Sciences, Guilin University of Electronic Technology, Guilin, 541004, China

    QiHuai Liu

  3. Department of Mathematics, Shanghai University, Shanghai, 200444, China

    XinXiang Li

Authors
  1. QiHuai Liu
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  2. XinXiang Li
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  3. Jun Yan
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Correspondence to QiHuai Liu.

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Liu, Q., Li, X. & Yan, J. Large time behavior of solutions for a class of time-dependent Hamilton-Jacobi equations. Sci. China Math. 59, 875–890 (2016). https://doi.org/10.1007/s11425-015-5102-5

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  • DOI: https://doi.org/10.1007/s11425-015-5102-5

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