Large Time Asymptotics of Hamilton–Jacobi Equations

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:

$$\displaystyle{p\mapsto H(x,p)\ \text{is convex},H(x,p) \geq H(x,0)\ \text{for all}\ (x,p)\,\in \,\mathbb{T}^{n} \times \mathbb{R}^{n},\ \max _{ x\,\in \,\mathbb{T}^{n}}H(x,0) = 0.}$$

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,

$$\displaystyle{u(x,t) - (v(x) - ct) \rightarrow 0\quad \text{uniformly for}\ x\,\in \,\mathbb{T}^{n},}$$

where u is the solution of the initial value problem and (v, c) is a solution to the associated ergodic problem.