On almost periodic viscosity solutions to Hamilton-Jacobi equations

TL;DR

This paper proves that almost periodic viscosity solutions to multidimensional Hamilton-Jacobi equations preserve their almost periodicity over time and explores decay and convergence properties in specific cases.

Contribution

It establishes the preservation of almost periodicity for viscosity solutions and analyzes their long-term behavior in multidimensional and one-dimensional settings.

Findings

Viscosity solutions remain spatially almost periodic over time.

In one dimension, solutions decay as time approaches infinity.

Periodic solutions converge to traveling waves.

Abstract

We establish that a viscosity solution to a multidimensional Hamilton-Jacobi equation with Bohr almost periodic initial data remains to be spatially almost periodic and the additive subgroup generated by its spectrum does not increase in time. In the case of one space variable and a non-degenerate hamiltonian we prove the decay property of almost periodic viscosity solutions when time $t\to+\infty$. For periodic solutions the more general result is proved on unconditional asymptotic convergence of a viscosity solution to a traveling wave.