Time periodic solutions of Hamilton-Jacobi equations with autonomous Hamiltonian on the circle.

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) ∈ S × [ 0 , + ∞) , where S is the unit circle and H = H (x , p , u) satisfies Tonelli conditions with respect to the argument p and is strictly decreasing in the argument u. We also study the long time behavior of viscosity solutions of the Cauchy problem for the above contact Hamilton-Jacobi equation. It is shown that for a class of initial data the corresponding viscosity solutions converge to asymptotic time periodic viscosity solutions. As an application of the existence result we analyze a bifurcation phenomenon for ∂ t w (x , t) + H (x , ∂ x w (x , t) , λ w (x , t)) = 0 where λ is a parameter. [ABSTRACT FROM AUTHOR]