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

2021/12/30 by Wang, Kaizhi, Yan, Jun, Zhao, Kai
#35D40 #35F21 #37J50 #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2112.14896

Abstract

We are concerned with the existence and multiplicity of nontrivial time-periodic viscosity solutions to ∂_t w(x,t) + H( x,∂_x w(x,t),w(x,t) )=0, (x,t)∈ \mathbbS × [0,+∞). We find that there are infinitely many nontrivial time-periodic viscosity solutions with different periods when (∂ H)/(∂ u)(x,p,u)\leqslant-δ<0 by analyzing the asymptotic behavior of the dynamical system (C(\mathbbS ,ℝ),\T_t\_{t\geqslant 0}), where \T_t\_{t\geqslant 0} was introduced in \citeWWY1. Moreover, in view of the convergence of T_tnφ, 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 ∂_t w(x,t) + H( x,∂_x w(x,t),λw(x,t) )=0, (x,t)∈ \mathbbS × [0,+∞), when the sign of the parameter λ varies. The structure of the unit circle \mathbbS plays an essential role here. The most important novelty is the discovery of the nontrivial recurrence of (C(\mathbbS ,ℝ),\T_t\_{t\geqslant 0}).

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