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

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.