Nonlinear perturbations of a class of holomorphic semigroups of growth order αα by comparison theorems for Volterra equations

This paper studies nonlinear perturbation of a holomorphic semigroup of growth order αα in a Banach space XX. The generator AA of a holomorphic semigroup is assumed to be almost sectorial and it is also assumed that the part A0A0 in the closure X0X0 of the domain D(A)D(A) is sectorial. A nonlinear perturbing operator BB is assumed to be locally continuous from a subset of a real interpolation space between X0X0 and D(A0)D(A0) into XX. 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.