Exponential convergence of 1-graph of the solution semigroup of contact Hamilton-Jacobi equations

TL;DR

This paper proves that the 1-graph of the solution semigroup for contact Hamilton-Jacobi equations converges exponentially to the stationary solution's 1-graph, highlighting differences between dissipative and conservative systems.

Contribution

It establishes exponential convergence of the solution semigroup's 1-graph in contact Hamilton-Jacobi equations, a novel result in weak KAM theory.

Findings

Exponential convergence of the 1-graph to stationary solutions

Difference between dissipative and conservative systems in weak KAM context

Convergence measured in Hausdorff metrics

Abstract

Under certain assumptions, we show that for the solution semigroup of evolutionary contact Hamilton-Jacobi equations, its 1-graph, as a pseudo Legendrian graph, converges exponentially to the 1-graph of the viscosity solution of stationary equations in the sense of certain Hausdorff metrics. This result reveals an essential difference between certain dissipative systems and conservative systems from weak KAM aspects.