Nonlinear perturbations of a class of holomorphic semigroups of growth order [formula omitted] by comparison theorems for Volterra equations

Abstract

This paper studies nonlinear perturbation of a holomorphic semigroup of growth order α in a Banach space X. The generator A of a holomorphic semigroup is assumed to be almost sectorial and it is also assumed that the part A0 in the closure X0 of the domain D(A) is sectorial. A nonlinear perturbing operator B is assumed to be locally continuous from a subset of a real interpolation space between X0 and D(A0) into X. Existence and uniqueness of mild solutions to the associated Cauchy problems are proved by using comparison theorems for integral equations of Volterra type. The result obtained is applied to show the global well-posedness of drift–diffusion systems with no-flux boundary conditions in the two dimensional case.