Nonlinear perturbations of evolution systems in scales of Banach spaces

TL;DR

This paper extends the abstract Cauchy-Kovalevskaya theorem to nonlinear evolution systems in Banach space scales, proving existence, uniqueness, and continuous dependence of solutions, with applications to biological models and nonlinear Fokker-Planck equations.

Contribution

It introduces a novel framework for solving nonlinear, non-autonomous evolution equations in Banach space scales, including new existence and uniqueness results.

Findings

Proved existence and uniqueness of solutions in Banach space scales.

Established continuous dependence of solutions on data and operators.

Applied results to biological and particle system models.

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 \[ \frac{du(t)}{dt} = A(t)u(t) + B(u(t),t), \ \ u(0) = x \] 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.