Linear evolution equations in scales of Banach spaces

TL;DR

This paper investigates linear evolution equations in Banach space scales, establishing existence, uniqueness, and stability of solutions, and applies these results to differential systems and the Fokker-Planck equation in continuum models.

Contribution

It introduces new analytical results for linear evolution equations in Banach scales, including dual problem analysis and applications to biological and physical models.

Findings

Proved existence and uniqueness of classical solutions.

Established stability in Banach space scales.

Applied results to Fokker-Planck and logistic models.

Abstract

This work is devoted to the study of a class of linear time-inhomogeneous evolution equations in a scale of Banach spaces. Existence, uniquenss and stability for classical solutions is provided. We study also the associated dual Cauchy problem for which we prove uniqueness in the dual scale of Banach spaces. The results are applied to an infinite system of ordinary differential equations but also to the Fokker-Planck equation associated with the spatial logistic model in the continuum.