Minimal periods for semilinear parabolic equations

We show that, if \(-A\) generates a bounded holomorphic semigroup in a Banach space X, \(\alpha \in [0,1)\), and \(f:D(A)\rightarrow X\) satisfies \(\Vert f(x)-f(y)\Vert \le L\Vert A^\alpha (x-y)\Vert \), then a non-constant T-periodic solution of the equation \({\dot{u}}+Au=f(u)\) satisfies \(LT^{1-\alpha }\ge K_\alpha \) where \(K_\alpha >0\) is a constant depending on \(\alpha \) and the semigroup. This extends results by Robinson and Vidal-Lopez, which have been shown for self-adjoint operators \(A\ge 0\) in a Hilbert space. For the latter case, we obtain - with a conceptually new proof - the optimal constant \(K_\alpha \), which only depends on \(\alpha \), and we also include the case \(\alpha =1\). In Hilbert spaces H and for \(\alpha =0\), we present a similar result with optimal constant where Au in the equation is replaced by a possibly unbounded gradient term \(\nabla _H{\mathscr {E}}(u)\). This is inspired by applications with bounded gradient terms in a paper by Mawhin and Walter.