Matrix parabolic problems in Sobolev spaces of generalized smoothness

TL;DR

This paper investigates linear parabolic differential systems in advanced Sobolev spaces, establishing conditions for solution regularity and continuity of derivatives using generalized smoothness characterized by slowly varying functions.

Contribution

It introduces a framework for analyzing parabolic problems in Sobolev spaces of generalized smoothness, proving topological isomorphisms and regularity conditions.

Findings

Proves topological isomorphisms for the parabolic problem in these spaces.

Provides necessary and sufficient conditions for solution regularity.

Derives exact conditions for the continuity of generalized derivatives.

Abstract

We study a general linear parabolic problem for Petrovskii parabolic differential system in Sobolev anisotropic distribution spaces of generalized smoothness. Slowly varying functions are used to characterize supplementary generalized smoothness that cannot be determined by number indexes. We prove that this problem induces topological isomorphisms on appropriate pairs of such spaces. As an application, we give sufficient and necessary conditions for the problem solutions to have prescribed generalized regularity expressed in terms of these spaces. Their use allows obtaining exact conditions for indicated generalized partial derivatives of the solutions to be continuous.