Time periodic and almost periodic viscosity solutions of contact Hamilton-Jacobi equations on ${\mathbb{T}}^n$

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 ${\mathbb{T}}^n$. 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 $[0,+\mathrm{\infty })$. Moreover, we clarify the structure of $D$. As a consequence, we also study the existence of Bohr almost periodic viscosity solutions.