Minimal periods for semilinear parabolic equations

We show that, if $$-A$$ - A generates a bounded holomorphic semigroup in a Banach space X , α ∈ [0,1) α ∈ [ 0 , 1 ) , and f:D(A)→ X f : D ( A ) → X satisfies f(x)-f(y) ≤ L A^α (x-y) ‖ f ( x ) - f ( y ) ‖ ≤ L ‖ A α ( x - y ) ‖ , then a non-constant T -periodic solution of the equation u̇+Au=f(u) u ˙ + A u = f ( u ) satisfies LT1-α≥ K_α L T 1 - α ≥ K α where K_α >0 K α > 0 is a constant depending on α α and the semigroup. This extends results by Robinson and Vidal-Lopez, which have been shown for self-adjoint operators A≥ 0 A ≥ 0 in a Hilbert space. For the latter case, we obtain - with a conceptually new proof - the optimal constant K_α K α , which only depends on α α , and we also include the case α =1 α = 1 . In Hilbert spaces H and for α =0 α = 0 , we present a similar result with optimal constant where Au in the equation is replaced by a possibly unbounded gradient term ∇ HE(u) ∇ H E ( u ) . This is inspired by applications with bounded gradient terms in a paper by Mawhin and Walter.