### Abstract

We study the long time behavior of viscosity solutions of the Cauchy problem for Hamilton-Jacobi equations in ℝ^{n}. We prove that if the Hamiltonian H(x, p) is coercive and strictly convex in a mild sense in p and upper semi-periodic in x, then any solution of the Cauchy problem "converges" to an asymptotic solution for any lower semi-almost periodic initial function.

Original language | English |
---|---|

Pages (from-to) | 784-807 |

Number of pages | 24 |

Journal | Communications in Partial Differential Equations |

Volume | 33 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2008 May |

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### Keywords

- Almost periodic functions
- Hamilton-Jacobi equations
- Long time behavior
- Weak KAM theory

### ASJC Scopus subject areas

- Mathematics(all)
- Analysis
- Applied Mathematics

### Cite this

*Communications in Partial Differential Equations*,

*33*(5), 784-807. https://doi.org/10.1080/03605300701257427

**Asymptotic solutions of Hamilton-Jacobi equations with semi-periodic Hamiltonians.** / Ichihara, Naoyuki; Ishii, Hitoshi.

Research output: Contribution to journal › Article

*Communications in Partial Differential Equations*, vol. 33, no. 5, pp. 784-807. https://doi.org/10.1080/03605300701257427

}

TY - JOUR

T1 - Asymptotic solutions of Hamilton-Jacobi equations with semi-periodic Hamiltonians

AU - Ichihara, Naoyuki

AU - Ishii, Hitoshi

PY - 2008/5

Y1 - 2008/5

N2 - We study the long time behavior of viscosity solutions of the Cauchy problem for Hamilton-Jacobi equations in ℝn. We prove that if the Hamiltonian H(x, p) is coercive and strictly convex in a mild sense in p and upper semi-periodic in x, then any solution of the Cauchy problem "converges" to an asymptotic solution for any lower semi-almost periodic initial function.

AB - We study the long time behavior of viscosity solutions of the Cauchy problem for Hamilton-Jacobi equations in ℝn. We prove that if the Hamiltonian H(x, p) is coercive and strictly convex in a mild sense in p and upper semi-periodic in x, then any solution of the Cauchy problem "converges" to an asymptotic solution for any lower semi-almost periodic initial function.

KW - Almost periodic functions

KW - Hamilton-Jacobi equations

KW - Long time behavior

KW - Weak KAM theory

UR - http://www.scopus.com/inward/record.url?scp=45849142961&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=45849142961&partnerID=8YFLogxK

U2 - 10.1080/03605300701257427

DO - 10.1080/03605300701257427

M3 - Article

VL - 33

SP - 784

EP - 807

JO - Communications in Partial Differential Equations

JF - Communications in Partial Differential Equations

SN - 0360-5302

IS - 5

ER -