Abstract
In the last decade, a number of authors have studied extensively the large time behavior of solutions of first-order Hamilton–Jacobi equations. Several convergence results have been established. The first general theorem in this direction was proven by Namah and Roquejoffre in [37], under the assumptions:
We will first discuss this setting in Sect. 5.2. In this setting, as the Hamiltonian has a simple structure, we are able to find an explicit subset of \(\mathbb{T}^{n}\) which has the monotonicity of solutions and the property of the uniqueness set. Therefore, we can relatively easily get a convergence result of the type (4.5), that is,
where u is the solution of the initial value problem and (v, c) is a solution to the associated ergodic problem.
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Le, N.Q., Mitake, H., Tran, H.V. (2017). Large Time Asymptotics of Hamilton–Jacobi Equations. In: Mitake, H., Tran, H. (eds) Dynamical and Geometric Aspects of Hamilton-Jacobi and Linearized Monge-Ampère Equations. Lecture Notes in Mathematics, vol 2183. Springer, Cham. https://doi.org/10.1007/978-3-319-54208-9_5
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