Nonlinear perturbations of evolution systems in scales of Banach spaces

Abstract

A variant of the abstract Cauchy-Kovalevskaya theorem is considered. We prove existence and uniqueness of classical solutions to the nonlinear, nonautonomous initial value problem du(t) dt = A(t)u(t) + B(u(t), t), u(0) = x in a scale of Banach spaces. Here A(t) is the generator of an evolution system acting in a scale of Banach spaces and B(u, t) obeys an Ovcyannikov-type bound. Continuous dependence of the solution with respect to A(t), B(u, t) and x is proved. The results are applied to the Kimura-Maruyama equation for the mutation-selection balance model. This yields a new insight in the construction and uniqueness question for nonlinear Fokker-Planck equations related with interacting particle systems in the continuum.