Some problems for Petrovskii parabolic systems in generalized Sobolev spaces

TL;DR

This paper investigates the regularity of solutions to Petrovskii parabolic systems in generalized Sobolev spaces, establishing isomorphisms and providing a finer characterization of solution regularity using function parameters.

Contribution

It introduces a framework for analyzing Petrovskii systems in anisotropic generalized Sobolev spaces with a function parameter for enhanced regularity characterization.

Findings

Proves isomorphisms between inhomogeneous initial-boundary value problems and generalized Sobolev spaces.

Characterizes solution regularity more precisely using a function parameter.

Extends classical Sobolev space analysis to a more flexible generalized setting.

Abstract

We consider an inhomogeneous initial-boundary value problem for a Petrovskii parabolic system of second order PDEs. We prove that this problem induces isomorphisms between appropriate anisotropic generalized Sobolev spaces. The regularity of these spaces are given by a pair of real numbers and by a function parameter. The latter allows us to characterize the regularity of solutions to the problem more finely as compared with anisotropic Sobolev spaces.