Sobre el problema de Cauchy para la ecuación del calor fraccionaria

Abstract

The Cauchy problem for the fractional heat equation (PF) is characterized by the fractional laplacian operator instead of the classical one which appears in the diffusion term of the well-known heat equation (PC). This article deals with two essential properties of the problem (PF): well-posedness in the Hadamard’s sense and maximal regularity of its mild solutions. This latter property has as motivation the result of maximal regularity of the heat-kernel known for (PC). In 2018, G. Grubb in her article: Regularity in Lp Sobolev Spaces of Solutions to Fractional Heat Equations, used the pseudo-differential operator theory to study the maximal regularity (in space and time) of weak solutions of the fractional heat equation in non-homogeneous anisotropic Sobolev spaces. In our work, we will study the maximal regularity (in space) of the mild solutions of (PF), using tools of harmonic analysis such as: the Fourier transform and the Sobolev spaces. Moreover, using the fact that the mild solution writes down with an explicity integral formulation as the sum of two terms in which the problem data of (PF) intervene separately, our main contribution will be an analysis of the maximal regularity that reaches each of these terms in the fractional Sobolev spaces.