Periodic and almost periodic evolution flows and their stability on non-compact Einstein manifolds and applications

Considering the evolution equations of parabolic type on a non-compact Einstein manifold $(\mathcal M,g)$ with negative Ricci curvature tensor, we establish results on the existence, uniqueness of time-periodic (on the time half-axis) and almost periodic (on the whole time axis) mild solutions to such equations. We develop a general framework for evolution equations on $(\mathcal M,g)$. Using certain dispersive and smoothing properties of semigroups, we construct a bounded (in time) mild solution and prove a Massera-type theorem for the linearized equations on the half-line using a mean-ergodic method to obtain the existence and uniqueness of periodic {and almost periodic solutions} to evolution equations. Next, using fixed point arguments, we can pass from linear equations to semilinear equations to prove the existence, uniqueness and stability of periodic solutions. Lastly, we apply our abstract results to establish the existence and stability of periodic and almost periodic solutions to Stokes and Navier–Stokes equations as well as to semilinear vectorial heat equations with rough coefficients on $(\mathcal M,g)$.