Dirichlet problems associated to abstract nonlocal space–time differential operators

Abstract

Let the abstract fractional space–time operator (∂t+A)s be given, where s∈(0,∞) and -A:D(A)⊆X→X is a linear operator generating a uniformly bounded strongly measurable semigroup (S(t))t≥0 on a complex Banach space X. We consider the corresponding Dirichlet problem of finding u:R→X such that (Formula presented.) for given t0∈R and g:(-∞,t0]→X. We define the concept of Lp-solutions, to which we associate a mild solution formula which expresses u in terms of g and (S(t))t≥0 and generalizes the well-known variation of constants formula for the mild solution to the abstract Cauchy problem u′+Au=0 on (t0,∞) with u(t0)=x∈D(A)¯. Moreover, we include a comparison to analogous solution concepts arising from Riemann–Liouville and Caputo type initial value problems.