ASYMPTOTIC SOLUTIONS FOR LARGE-TIME OF HAMILTON-JACOBI EQUATIONS IN EUCLIDEAN $n$ SPACE(Viscosity Solution Theory of Differential Equations and its Developments)

We study the large time behavior of solutions of the Cauchy problem for the Hamilton–Jacobi equation ut+H(x,Du)=0 in Rn×(0,∞), where H(x,p) is continuous on Rn×Rn and convex in p. We establish a general convergence result for viscosity solutions u(x,t) of the Cauchy problem as t→∞.

[1]  J. Roquejoffre,et al.  Remarks on the long time behaviour of the solutions of hamilton-jacobi equations , 1999 .

[2]  P. Lions,et al.  Uniqueness of viscosity solutions of Hamilton-Jacobi equations revisited , 1987 .

[3]  Hiroyoshi Mitake.  Asymptotic Solutions of Hamilton-Jacobi Equations with State Constraints , 2007 .

[4]  E. Barron,et al.  OPTIMAL CONTROL AND SEMICONTINUOUS VISCOSITY SOLUTIONS , 1991 .

[5]  P. Lions.  Generalized Solutions of Hamilton-Jacobi Equations , 1982 .

[6]  H. Ishii.  A simple, direct proof of uniqueness for solutions of the hamilton-jacobi equations of eikonal type , 1987 .

[7]  Hitoshi Ishii,et al.  Asymptotic solutions of Hamilton-Jacobi equations in Euclidean n space , 2006 .

[8]  P. Lions,et al.  User’s guide to viscosity solutions of second order partial differential equations , 1992, math/9207212.

[9]  Guy Barles,et al.  On the Large Time Behavior of Solutions of Hamilton-Jacobi Equations , 2000, SIAM J. Math. Anal..

[10]  Guy Barles.  Asymptotic behavior of viscosity solutions of first-order Hamilton Jacobi equations , 1985 .

[11]  Guy Barles,et al.  Space-Time Periodic Solutions and Long-Time Behavior of Solutions to Quasi-linear Parabolic Equations , 2001, SIAM J. Math. Anal..

[12]  A. Fathi,et al.  Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens , 1997 .

[13]  Antonio Siconolfi,et al.  PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians , 2005 .

[14]  J. Aubin,et al.  Differential inclusions set-valued maps and viability theory , 1984 .

[15]  H. Ishii,et al.  Asymptotic Solutions of Hamilton–Jacobi Equations with Semi-Periodic Hamiltonians , 2008 .

[16]  Guy Barles,et al.  Ergodic Type Problems and Large Time Behaviour of Unbounded Solutions of Hamilton–Jacobi Equations , 2006, math/0612704.

[17]  石井 仁司.  Asymptotic solutions for large-time of Hamilton-Jacobi equations in Euclidean n space (微分方程式の粘性解理論とその発展--RIMS研究集会報告集) , 2007 .

[18]  H. Ishii,et al.  Asymptotic Solutions of Viscous Hamilton–Jacobi Equations with Ornstein–Uhlenbeck Operator , 2006 .

[19]  H. Ishii.  Comparison results for hamilton-jacobi equations without grwoth condition on solutions from above , 1997 .

[20]  J. Ball.  GENERALIZED SOLUTIONS OF HAMILTON-JACOBI EQUATIONS (Research Notes in Mathematics, 69) , 1983 .

[21]  E. Barron,et al.  Semicontinuous Viscosity Solutions For Hamilton–Jacobi Equations With Convex Hamiltonians , 1990 .

[22]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[23]  M. Bardi,et al.  Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations , 1997 .

[24]  Andrea Davini,et al.  A Generalized Dynamical Approach to the Large Time Behavior of Solutions of Hamilton-Jacobi Equations , 2006, SIAM J. Math. Anal..

[25]  L. Evans,et al.  A survey of partial differential equations methods in weak KAM theory , 2004 .

[26]  Albert Fathi,et al.  Weak KAM Theorem in Lagrangian Dynamics , 2001 .

[27]  Hitoshi Ishii,et al.  Representation formulas for solutions of Hamilton-Jacobi equations with convex Hamiltonians , 2007 .

[28]  Olivier Alvarez,et al.  Bounded-from-below viscosity solutions of Hamilton-Jacobi equations , 1997, Differential and Integral Equations.

[29]  Antonio Siconolfi,et al.  Existence of C1 critical subsolutions of the Hamilton-Jacobi equation , 2004 .

[30]  G. Barles.  Solutions de viscosité des équations de Hamilton-Jacobi , 1994 .

[31]  H. Ishii.  A generalization of a theorem of Barron and Jensen and a comparison theorem for lower semicontinuous viscosity solutions , 2001, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[32]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[33]  S. Zagatti.  On viscosity solutions of Hamilton-Jacobi equations , 2008 .

[34]  Jean-Michel Roquejoffre,et al.  Convergence to steady states or periodic solutions in a class of Hamilton–Jacobi equations , 2001 .

[35]  L. Evans.  Periodic homogenisation of certain fully nonlinear partial differential equations , 1992, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[36]  A. Fathi.  Sur la convergence du semi-groupe de Lax-Oleinik , 1998 .