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.

中文翻译：

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Banach空间尺度演化系统的非线性扰动

考虑了抽象 Cauchy-Kovalevskaya 定理的一个变体。我们证明了非线性、非自治初值问题的经典解的存在性和唯一性 \[ \frac{du(t)}{dt} = A(t)u(t) + B(u(t),t) , \ \ u(0) = x \] 在巴拿赫空间的尺度上。这里 $A(t)$ 是在 Banach 空间尺度上作用的演化系统的生成器，$B(u,t)$ 服从 Ovcyannikov 型边界。证明了解对 $A(t)$、$B(u,t)$ 和 $x$ 的连续依赖性。结果应用于突变-选择平衡模型的木村-丸山方程。这对与连续介质中相互作用的粒子系统相关的非线性 Fokker-Planck 方程的构造和唯一性问题产生了新的见解。