Existence and uniqueness properties for solutions of a class of Banach space valued evolution equations

TL;DR

This paper presents a comprehensive and self-contained proof of existence and uniqueness for a broad class of Banach space valued evolution equations, encompassing ODEs, PDEs, and certain SPDEs, under mild regularity conditions.

Contribution

It offers a general existence and uniqueness theorem for Banach space evolution equations with additive forcing, including nonlinear semigroups, with a self-contained proof.

Findings

Covers finite-dimensional ODEs and infinite-dimensional PDEs and SPDEs

Allows nonlinear semigroup operators under mild regularity

Provides a unified, self-contained proof of existence and uniqueness

Abstract

In this note we provide a self-contained proof of an existence and uniqueness result for a class of Banach space valued evolution equations with an additive forcing term. The framework of our abstract result includes, for example, finite dimensional ordinary differential equations (ODEs), semilinear deterministic partial differential equations (PDEs), as well as certain additive noise driven stochastic partial differential equations (SPDEs) as special cases. The framework of our general result assumes somehow mild regularity conditions on the involved semigroup and also allows the involved semigroup operators to be nonlinear. The techniques used in the proofs of our results are essentially well-known in the relevant literature. The contribution of this note is to provide a rather general existence and uniqueness result which covers several situations as special cases and also to provide a self-contained proof for this existence and uniqueness result.