Time-periodic solutions of contact Hamilton-Jacobi equations on the circle

TL;DR

This paper investigates the existence, multiplicity, and long-term behavior of nontrivial time-periodic viscosity solutions to contact Hamilton-Jacobi equations on the circle, revealing infinitely many solutions with different periods and a bifurcation phenomenon influenced by a parameter.

Contribution

It introduces a novel analysis of the dynamical system associated with the equations, demonstrating nontrivial recurrence and bifurcation phenomena on the circle.

Findings

Infinitely many nontrivial time-periodic solutions with different periods exist.

Long-time convergence and recurrence properties of the solutions are established.

A bifurcation phenomenon depending on a parameter is described, highlighting the circle's structure.

Abstract

We are concerned with the existence and multiplicity of nontrivial time-periodic viscosity solutions to \[ \partial_t w(x,t) + H( x,\partial_x w(x,t),w(x,t) )=0,\quad (x,t)\in \mathbb{S} \times [0,+\infty). \] We find that there are infinitely many nontrivial time-periodic viscosity solutions with different periods when $\frac{\partial H}{\partial u}(x,p,u)\leqslant-\delta<0$ by analyzing the asymptotic behavior of the dynamical system $(C(\mathbb{S} ,\mathbb{R}),\{T_t\}_{t\geqslant 0})$, where $\{T_t\}_{t\geqslant 0}$ was introduced in \cite{WWY1}. Moreover, in view of the convergence of $T_{t_n}\varphi$, we get the existence of nontrivial periodic points of $T_t$, where $\varphi$ are initial data satisfying certain properties. This is a long-time behavior result for the solution to the above equation with initial data $\varphi$. At last, as an application, we describe to readers a bifurcation phenomenon for \[ \partial_t w(x,t) + H( x,\partial_x w(x,t),\lambda w(x,t) )=0,\quad (x,t)\in \mathbb{S} \times [0,+\infty), \] when the sign of the parameter $\lambda$ varies. The structure of the unit circle $\mathbb{S}$ plays an essential role here. The most important novelty is the discovery of the nontrivial recurrence of $(C(\mathbb{S} ,\mathbb{R}),\{T_t\}_{t\geqslant 0})$.