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A global bifurcation theorem for a positone multiparameter problem and its application
Long-time asymptotic solutions of convex hamilton-jacobi equations depending on unknown functions
Suzhou University of Science and Technology, Suzhou 215009, China |
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 $H(x, u(x), Du(x))= 0$.
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. |
show all references
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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