Long-time evolution of forced waves in the low viscosity regime

We consider a model for internal waves described by a zero order pseudo-differential Hamiltonian $P$ damped by a second order viscosity term $i\nu Q$. Under Morse-Smale or similar weaker global conditions on the classical dynamics, we describe qualitatively the long-time behavior of solutions of the corresponding evolution equation with smooth forcing in a small $\nu$ regime. We show that dissipation effects arise no earlier than at the $t\sim {\nu }^{-1\mathrm{/}3-}$ time scale.