Minimal periods for semilinear parabolic equations

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