Maximal regularity of linear vector-valued parabolic Volterra equations
The author investigates maximal regularity of type \(C^ \alpha\) for a Volterra equation \((*)\;u(t)=f(t)+\int^ t_ 0a(t-s)Au(s)ds\) in a Banach space X, where \(A\) generates a holomorphic semigroup; i.e., conditions are given under which for every \(f\in C^ \alpha_ 0\) there is a unique solution \(u\in C^ \alpha_ 0\) of \((*)\). Applications to viscoelasticity and heat conduction are given. The proofs are based on an interesting result about inversion of Laplace transforms: If \(g\) is holomorphic in the right half plane and \(\|\lambda g(\lambda)\|\leq M\), \(\|\lambda^ 2g'(\lambda)\|\leq M\), then \(g\) is the Laplace transform of a bounded function (if X is reflexive).
scientific article; zbMATH DE number 40686