EVOLUTION FLOWS ON NON-COMPACT RIEMANNIAN MANIFOLDS: STABILITY, PERIODICITY, AND APPLICATIONS

Considering the evolution equations on a non-compact Riemannian manifold (M, g) with negative Ricci curvature tensor and non-positive sectional curvature, we construct a general framework to establish the existence and stability of a bounded (in time) solution and prove a Serrin-type theorem on the existence of a time-periodic mild solution to such equations. In our strategy, we develop certain axioms on dispersive and smoothing properties of semigroups. Such properties allow us to derive necessary estimates to obtain the boundedness and stability of solutions, and to establish the Serrin-type theorem for evolution equations in Riemannian manifolds. Then, we apply our abstract results to establish the existence and stability of periodic solutions to Boussinesq systems on non-compact Riemannian manifolds with negative Ricci curvature tensor and non-positive sectional curvature.