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

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 $\frac{\partial H}{\partial u}(x,p,u)⩽-\delta <0$ by analyzing the asymptotic behavior of the dynamical system $(C(\mathbb{S},\mathbb{R}),\{T_t{\}}_{t⩾0})$, where $\{T_t{\}}_{t⩾0}$ was introduced in \cite{WWY1}. Moreover, in view of the convergence of $T_{t_n}\phi$, we get the existence of nontrivial periodic points of $T_t$, where $\phi$ are initial data satisfying certain properties. This is a long-time behavior result for the solution to the above equation with initial data $\phi$. At last, as an application, we describe to readers a bifurcation phenomenon for \[ \partial_t w(x,t) + H( x,\partial_x w(x,t),λw(x,t) )=0,\quad (x,t)\in \mathbb{S} \times [0,+\infty), \] 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 $(C(\mathbb{S},\mathbb{R}),\{T_t{\}}_{t⩾0})$.