Around Ovsyannikov's method

TL;DR

This paper extends Ovsyannikov's method to analyze the existence, uniqueness, and behavior of solutions for abstract linear evolution equations in Banach space scales, with applications to stochastic birth-death processes.

Contribution

It provides a generalized abstract framework for Ovsyannikov's method applicable to a broader class of evolution equations with an application to stochastic dynamics.

Findings

Established existence and uniqueness results for the evolution equations.

Derived limiting behavior of solutions in the Banach space scale.

Applied the abstract results to birth-and-death stochastic dynamics.

Abstract

We study existence, uniqueness, and a limiting behaviour of solutions to an abstract linear evolution equation in a scale of Banach spaces. The generator of the equation is a perturbation of the operator which satisfies the classical assumptions of Ovsyannikov's method by a generator of a C_0-semigroup acting in each of the spaces of the scale. The results are (slightly modified) abstract version of those considered in [Math. Models Methods Appl. Sci., 25, 2, 2015, pp.343-370] for a particular equation. An application to a birth-and-death stochastic dynamics in the continuum is considered.