A pde with drift of negative Besov index and linear growth solutions

TL;DR

This paper studies PDEs with coefficients in negative Besov spaces and solutions with linear growth, establishing existence, uniqueness, and continuity results using specialized Besov-Hölder spaces.

Contribution

It introduces new Besov-Hölder type spaces suitable for linear growth solutions and proves key properties of these spaces and the associated PDEs.

Findings

Existence and uniqueness of solutions in the new spaces

Equivalence of mild and weak solutions in this setting

Introduction of a separable subclass of Besov-Hölder spaces

Abstract

This paper investigates a class of PDEs with coefficients in negative Besov spaces and whose solutions have linear growth. We show existence and uniqueness of mild and weak solutions, which are equivalent in this setting, and several continuity results. To this aim, we introduce ad-hoc Besov-H{\"o}lder type spaces that allow for linear growth, and investigate the action of the heat semigroup on them. We conclude the paper by introducing a special subclass of these spaces which has the useful property to be separable.