Minimal periods for semilinear parabolic equations

Abstract

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