Minimal periods of semilinear evolution equations with Lipschitz nonlinearity revisited

We show that when A   is a self-adjoint sectorial operator on a Hilbert space, for 0⩽α<10⩽α<1 there exists a constant KαKα, depending only on α  , such that if f:D(Aα)→Xf:D(Aα)→X satisfies‖f(u)−f(v)‖X⩽L‖Aα(u−v)‖X‖f(u)−f(v)‖X⩽L‖Aα(u−v)‖X then any periodic orbit of the equation u˙=−Au+f(u) has period at least KαL−1/(1−α)KαL−1/(1−α). This generalises our previous result [J.C. Robinson, A. Vidal-López, Minimal periods of semilinear evolution equations with Lipschitz nonlinearity, J. Differential Equations 220 (2006) 396–406] which was restricted to 0⩽α⩽1/20⩽α⩽1/2 and A−1A−1 compact.