Time periodic solutions of Hamilton-Jacobi equations with autonomous Hamiltonian on the circle

We are concerned with the existence and multiplicity of nontrivial time periodic viscosity solutions to partial differential tw(x, t) + H(x, partial differential xw(x, t), w(x, t)) = 0, (x, t) E S x [0, +oo), where S is the unit circle and H = H(x, p, u) satisfies Tonelli conditions with respect to the argument p and is strictly decreasing in the argument u. We also study the long time behavior of viscosity solutions of the Cauchy problem for the above contact Hamilton-Jacobi equation. It is shown that for a class of initial data the corresponding viscosity solutions converge to asymptotic time periodic viscosity solutions. As an application of the existence result we analyze a bifurcation phenomenon for partial differential tw(x, t) + H(x, partial differential xw(x, t), lambda w(x, t)) = 0 where lambda is a parameter.(c) 2023 Elsevier Masson SAS. All rights reserved.

Automated summary

This article discusses the existence and multiplicity of nontrivial time periodic viscosity solutions to a contact Hamilton-Jacobi equation, and investigates the long time behavior of these solutions. It is found that for a certain class of initial data, the corresponding viscosity solutions converge to asymptotic time periodic viscosity solutions. The article also analyzes a bifurcation phenomenon for a parameter-dependent Hamilton-Jacobi equation.