Minimal periods of semilinear evolution equations with Lipschitz nonlinearity

It is known that any periodic orbit of a Lipschitz ordinary differential equation ẋ = f(x) must have period at least 2π/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 α with 0 ≤ α ≤ 1/2 there exists a constant K_α such that if L is the Lipschitz constant of f as a map from D(A^α) into H then any periodic orbit has period at least K_αL^-1/(1-α). 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.