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

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 $\frac{\partial H}{\partial u}(x,p,u)\leqslant-δ<0$ 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 $T_{t_n}φ$, we get the existence of nontrivial periodic points of $T_t$, where $φ$ are initial data satisfying certain properties. This is a long-time behavior result for the solution to the above equation with initial data $φ$. 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 $λ$ 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 $(C(\mathbb{S} ,\mathbb{R}),\{T_t\}_{t\geqslant 0})$.

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