Minimal periods of semilinear evolution equations with Lipschitz nonlinearity

It is known that any periodic orbit of a Lipschitz ordinary differential equation x = f(x) must have period at least 2 pi/L, where L is the Lipschitz constant of f. In this paper, we prove a similar result for the semilinear evolution equation du/dt = -Au + f(u): for each alpha with 0 <= alpha <= 1/2 there exists a constant K-alpha such that if L is the Lipschitz constant of f as a map from D(A(alpha)) into H then any periodic orbit has period at least K alpha L-1/(1-alpha). As a concrete application we recover a result of Kukavica giving a lower bound on the period for the 2d Navier-Stokes equations with periodic boundary conditions. (c) 2005 Elsevier Inc. All rights reserved.