Time periodic and almost periodic viscosity solutions of contact Hamilton-Jacobi equations on Tn

This paper concerns with the time periodic viscosity solution problem for a class of evolutionary contact Hamilton-Jacobi equations with time independent Hamiltonians on the torus $T^n$. Under certain suitable assumptions we show that the equation has a non-trivial T-periodic viscosity solution if and only if $T\in D$, where D is a dense subset of $[0,+\infty )$. Moreover, we clarify the structure of D. As a consequence, we also study the existence of Bohr almost periodic viscosity solutions.