Cauchy problem for a fractional anisotropic parabolic equation in anisotropic H\"{o}lder spaces

TL;DR

This paper studies a generalized fractional anisotropic parabolic equation in anisotropic Hölder spaces, establishing conditions for the operator's isomorphism and unique solvability of the Cauchy problem.

Contribution

It extends the heat equation to fractional anisotropic cases with different powers for space variables and analyzes solvability in anisotropic Hölder spaces.

Findings

Operator is an isomorphism under certain conditions.

Unique solvability of the Cauchy problem proven.

Conditions on the fractional order of time derivative established.

Abstract

We consider a Cauchy problem for a fractional anisotropic parabolic equation in anisotropic H\"{o}lder spaces. The equation generalizes the heat equation to the case of fractional power of the Laplace operator and the power of this operator can be different with respect to different groups of space variables. The time derivative can be either fractional Caputo - Jrbashyan derivative or usual derivative. Under some necessary conditions on the order of the time derivative we show that the operator of the whole problem is an isomorphism of appropriate anisotropic H\"{o}lder spaces. Under some another conditions we prove unique solvability of the Cauchy problem in the same spaces. The final version of this paper is published by AIMS Evolution Equations and Control Theory at https://www.aimsciences.org/article/doi/10.3934/eect.2022029, doi: 10.3934/eect.2022029.