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

Naoyuki Ichihara; Hitoshi Ishii

doi:10.1080/03605300701257427

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

Naoyuki Ichihara, Hitoshi Ishii^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

22 Citations (Scopus)

Abstract

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.

Original language	English
Pages (from-to)	784-807
Number of pages	24
Journal	Communications in Partial Differential Equations
Volume	33
Issue number	5
DOIs	https://doi.org/10.1080/03605300701257427
Publication status	Published - 2008 May

Keywords

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

ASJC Scopus subject areas

Mathematics(all)
Analysis
Applied Mathematics

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10.1080/03605300701257427

Cite this

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author = "Naoyuki Ichihara and Hitoshi Ishii",

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AU - Ichihara, Naoyuki

AU - Ishii, Hitoshi

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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

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JO - Communications in Partial Differential Equations

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ER -

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

Abstract

Keywords

ASJC Scopus subject areas

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