Viscosity solutions of contact Hamilton-Jacobi equations with Hamiltonians depending periodically on unknown functions

Abstract

Assume $ H = H(x, p, u) $ with $ (x, p)\in T^*M $ and $ u\in\mathbb{S} $, is smooth and satisfies Tonelli conditions in $ p $, Lipschitz continuity condition in $ u $, where $ M $ is a compact connected smooth manifold without boundary. We find a compact interval $ \mathcal C $ such that equation \begin{document}$ H(x, \partial_xu(x), u(x)) = c $\end{document} has solutions if and only if $ c\in \mathcal C $. We also study the long-time behavior of the unique viscosity solution $ u^c $ of$ \partial_t u(x, t)+H(x, \partial_xu(x, t), u(x, t)) = c, \quad u(x, 0) = \varphi(x)\in C(M, \mathbb{R}). $If $ c\in \mathcal C $, $ u^c $ is bounded by a constant independent of $ c $ and Lipschitz with respect to the argument $ x $ with a Lipschitz constant independent of $ c $ and $ \varphi $. If $ c\notin \mathcal C $, then the long-time average of $ u^c $ can be characterized by a function $ c\mapsto\rho(c) $ which admits a modulus of continuity. We obtain these results by analyzing properties of a kind of one-parameter semigroups of operators. All the aforementioned results show the fundamental difference between Hamilton-Jacobi equations with Hamiltonians $ H(x, p, u) $ and $ \bar H(x, p) $.