Abstract
A variant of the abstract Cauchy–Kovalevskaya theorem is considered. We prove existence and uniqueness of classical solutions to the nonlinear, non-autonomous initial value problem 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.
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