Strong solution for singularly nonautonomous evolution equation with almost sectorial operators

Abstract

ct. In this paper we consider the singularly nonautonomous evolution problem ut + A(t)u = f(t), τ < t < τ + T; u(τ) = u0 ∈ X, associated with a family of uniformly almost sectorial linear operators A(t) : D ⊂ X → X, that is, a family for which a sector of the complex plane is contained in the resolvent of −A(t) and satisfies k(λ + A(t))−1kL(X) ≤ C |λ|α , for some α ∈ (0, 1), uniformly in t. Under a proper condition on the value of α we prove that the linear process associated to the family A(t), t ∈ R, is strongly differentiable and that the singularly nonautonomous problem has a strong solution. An example of a singularly nonautonomous reaction-diffusion equation in a domain with a handle illustrates the abstracts results obtai