Long-time asymptotic solutions of convex hamilton-jacobi equations depending on unknown functions

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October 2017, 37(10): 5151-5162. doi: 10.3934/dcds.2017223

Long-time asymptotic solutions of convex hamilton-jacobi equations depending on unknown functions

Suzhou University of Science and Technology, Suzhou 215009, China

* Corresponding author: Xia Li

Received September 2016 Revised April 2017 Published June 2017

Fund Project: The first author is supported by National Natural Science Foundation of China (Grant 11471238).

We study the long-time asymptotic behaviour of viscosity solutions $u(x,~t)$ of the Hamilton-Jacobi equation $u_t(x, t)+ H(x, u(x, t),$ $Du(x, t))= 0$ in $\mathbb{T}^n× {(-∞, ∞)}$, where $H= H(x, u, p)$ is convex and coercive in p and non-decreasing on u, and establish the uniform convergence of u to an an asymptotic solution u_∞ as $t~\to \text{ }\infty $. Moreover, u_∞ is a viscosity solution of Hamilton-Jacobi equation $H(x, u(x), Du(x))= 0$.

Keywords: Hamilton-Jacobi equations, viscosity solutions, long-time asymptotic solutions, PDE-viscosity solutions approach, dynamical approach.

Mathematics Subject Classification: Primary: 35D40, 35F21; Secondary: 49R99.

Citation: Xia Li. Long-time asymptotic solutions of convex hamilton-jacobi equations depending on unknown functions. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5151-5162. doi: 10.3934/dcds.2017223

References:

[1]	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. doi: 10.1007/s13373-013-0036-0.
[2]	G. Barles and P. E. Souganidis, On the large time behaviour of solutions of Hamilton-Jacobi equations, SIAM J. Math. Anal., 31 (2000), 925-939. doi: 10.1137/S0036141099350869.
[3]	P. Cannarsa and C. Sinestrari, Semiconcave Functions, H-J Equations, and Optimal Control, Prog. Nonlinear Differential Equations Appl., 58 (2004), Birkhäuser Boston, Inc., Boston, MA.
[4]	M. G. Crandall and P. L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), 1-42. doi: 10.1090/S0002-9947-1983-0690039-8.
[5]	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.
[6]	A. Davini and A. Siconolfi, A generalized dynamical approach to the large time behaviour of solutions of Hamilton-Jacobi equations, SIAM J. Math. Anal., 38 (2006), 478-502. doi: 10.1137/050621955.
[7]	A. Fathi, Théoréme KAM faible et théorie de Mather sur les systémes Lagrangiens, C. R. Acad. Sci. Paris Sér. I Math., 324 (1997), 1043-1046. doi: 10.1016/S0764-4442(97)87883-4.
[8]	A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik, C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 267-270. doi: 10.1016/S0764-4442(98)80144-4.
[9]	H. Ishii, Long-time asymptotic solutions of convex H-J equations with Neumann type boundary conditions, Calc. Var. Partial Differ. Equ., 42 (2011), 189-209. doi: 10.1007/s00526-010-0385-4.
[10]	H. Ishii, A short introduction to viscosity solutions and the large time behaviour of solutions of H-J equations, Lecture Notes in Mathematics, 2074 (2013), 111-249. doi: 10.1007/978-3-642-36433-4_3.
[11]	H. Ishii, Weak KAM aspects of convex Hamilton-Jacobi equations with Neumann type boundary conditions, J. Math. Pures Appl., 95 (2011), 99-135. doi: 10.1016/j.matpur.2010.10.006.
[12]	P. L. Lions, Generalized Solutions of Hamilton-Jacobi Equations Research notes in Mathematics, 69 (1982), Pitman (Advanced Publishing Program).
[13]	G. Namah and J. M. Roquejoffre, Remarks on the long time behaviour of the solutions ofHamilton-Jacobi equations, Comm. Partial Differential Equations, 24 (1999), 883-893. doi: 10.1080/03605309908821451.
[14]	J. M. Roquejoffre, Convergence to steady states or periodic solutions in a class of Hamilton-Jacobi equations, J. Math.Pures Appl., 80 (2001), 85-104. doi: 10.1016/S0021-7824(00)01183-1.
[15]	X. Su, L. Wang and J. Yan, Weak KAM theory for Hamilton-Jacobi equations depending on unknown functions, Discrete Contin. Dyn. Syst., 36 (2016), 6487-6522. doi: 10.3934/dcds.2016080.

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References:

[1]	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. doi: 10.1007/s13373-013-0036-0.
[2]	G. Barles and P. E. Souganidis, On the large time behaviour of solutions of Hamilton-Jacobi equations, SIAM J. Math. Anal., 31 (2000), 925-939. doi: 10.1137/S0036141099350869.
[3]	P. Cannarsa and C. Sinestrari, Semiconcave Functions, H-J Equations, and Optimal Control, Prog. Nonlinear Differential Equations Appl., 58 (2004), Birkhäuser Boston, Inc., Boston, MA.
[4]	M. G. Crandall and P. L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), 1-42. doi: 10.1090/S0002-9947-1983-0690039-8.
[5]	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.
[6]	A. Davini and A. Siconolfi, A generalized dynamical approach to the large time behaviour of solutions of Hamilton-Jacobi equations, SIAM J. Math. Anal., 38 (2006), 478-502. doi: 10.1137/050621955.
[7]	A. Fathi, Théoréme KAM faible et théorie de Mather sur les systémes Lagrangiens, C. R. Acad. Sci. Paris Sér. I Math., 324 (1997), 1043-1046. doi: 10.1016/S0764-4442(97)87883-4.
[8]	A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik, C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), 267-270. doi: 10.1016/S0764-4442(98)80144-4.
[9]	H. Ishii, Long-time asymptotic solutions of convex H-J equations with Neumann type boundary conditions, Calc. Var. Partial Differ. Equ., 42 (2011), 189-209. doi: 10.1007/s00526-010-0385-4.
[10]	H. Ishii, A short introduction to viscosity solutions and the large time behaviour of solutions of H-J equations, Lecture Notes in Mathematics, 2074 (2013), 111-249. doi: 10.1007/978-3-642-36433-4_3.
[11]	H. Ishii, Weak KAM aspects of convex Hamilton-Jacobi equations with Neumann type boundary conditions, J. Math. Pures Appl., 95 (2011), 99-135. doi: 10.1016/j.matpur.2010.10.006.
[12]	P. L. Lions, Generalized Solutions of Hamilton-Jacobi Equations Research notes in Mathematics, 69 (1982), Pitman (Advanced Publishing Program).
[13]	G. Namah and J. M. Roquejoffre, Remarks on the long time behaviour of the solutions ofHamilton-Jacobi equations, Comm. Partial Differential Equations, 24 (1999), 883-893. doi: 10.1080/03605309908821451.
[14]	J. M. Roquejoffre, Convergence to steady states or periodic solutions in a class of Hamilton-Jacobi equations, J. Math.Pures Appl., 80 (2001), 85-104. doi: 10.1016/S0021-7824(00)01183-1.
[15]	X. Su, L. Wang and J. Yan, Weak KAM theory for Hamilton-Jacobi equations depending on unknown functions, Discrete Contin. Dyn. Syst., 36 (2016), 6487-6522. doi: 10.3934/dcds.2016080.

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