On the Cauchy problem for integro-differential equations in the scale of spaces of generalized smoothness

TL;DR

This paper investigates the well-posedness of a parabolic integro-differential Cauchy problem within Lp-spaces characterized by generalized smoothness, establishing existence and uniqueness through a priori estimates and Levy process analysis.

Contribution

It introduces a framework for analyzing parabolic integro-differential equations in generalized smoothness spaces using Levy measures, providing new existence and uniqueness results.

Findings

Proved existence and uniqueness of solutions.

Derived a priori estimates for solutions.

Provided probability density estimates for Levy processes.

Abstract

Parabolic integro-differential model Cauchy problem is considered in the scale of Lp -spaces of functions whose regularity is defined by a scalable Levy measure. Existence and uniqueness of a solution is proved by deriving apriori estimates. Some rough probability density function estimates of the associated Levy process are used as well.