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

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.