VISCOSITY SOLUTIONS OF CONTACT HAMILTON-JACOBI EQUATIONS WITH HAMILTONIANS DEPENDING PERIODICALLY ON UNKNOWN FUNCTIONS

Assume H = H(x, p, u) with (x, p) is an element of T*M and u is an element of 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 C such that equationH(x, partial differential xu(x), u(x)) = chas solutions if and only if c is an element of C. We also study the long-time behavior of the unique viscosity solution uc of partial differential tu(x, t) + H(x, partial differential xu(x, t), u(x, t)) = c, u(x, 0) = phi(x) is an element of C(M, R).If c is an element of C, uc is bounded by a constant independent of c and Lipschitz with respect to the argument x with a Lipschitz constant independent of c and phi. If c is an element of/ C, then the long-time average of uc can be characterized by a function c 7 -> 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 over line H(x, p).