# Parabolic problems in generalized Sobolev spaces

## TL;DR

This paper studies the solvability and regularity of inhomogeneous parabolic boundary value problems within generalized anisotropic Sobolev spaces, introducing a function parameter to characterize subordinate regularity and establishing isomorphism results.

## Contribution

It introduces a framework for analyzing parabolic problems in generalized Sobolev spaces with a function parameter, proving operator isomorphism and regularity results.

## Key findings

- Operator is an isomorphism on generalized Sobolev spaces
- Established local regularity of solutions
- Derived conditions for continuity of generalized derivatives

## Abstract

We consider a general inhomogeneous parabolic initial-boundary value problem for a $2b$-parabolic differential equation given in a finite multidimensional cylinder. We investigate the solvability of this problem in some generalized anisotropic Sobolev spaces. They are parametrized with a pair of positive numbers $s$ and $s/(2b)$ and with a function $\varphi:[1,\infty)\to(0,\infty)$ that varies slowly at infinity. The function parameter $\varphi$ characterizes subordinate regularity of distributions with respect to the power regularity given by the number parameters. We prove that the operator corresponding to this problem is an isomorphism on appropriate pairs of these spaces. As an application, we give a theorem on the local regularity of the generalized solution to the problem. We also obtain sharp sufficient conditions under which chosen generalized derivatives of the solution are continuous on a given set.