On decay of almost periodic viscosity solutions to Hamilton-Jacobi equations

TL;DR

This paper proves that viscosity solutions to certain multidimensional Hamilton-Jacobi equations with almost periodic initial data decay to their minimum value over time.

Contribution

It establishes the decay behavior of viscosity solutions with almost periodic initial data for multidimensional Hamilton-Jacobi equations, extending understanding of long-term solution behavior.

Findings

Viscosity solutions decay to their infimum as time approaches infinity.

Decay occurs for equations with convex, non-degenerate Hamiltonians.

Results apply to multidimensional settings with Bohr almost periodic initial data.

Abstract

We establish that a viscosity solution to a multidimensional Hamilton-Jacobi equation with a convex non-degenerate hamiltonian and Bohr almost periodic initial data decays to its infimum as time $t\to+\infty$.