Time fractional parabolic equations with measurable coefficients and embeddings for fractional parabolic Sobolev spaces

TL;DR

This paper studies time fractional parabolic equations with measurable coefficients, establishing solvability in Sobolev spaces and providing new embeddings for fractional parabolic Sobolev spaces.

Contribution

It introduces novel solvability results for fractional parabolic equations with measurable coefficients and offers elementary proofs for embeddings in fractional Sobolev spaces.

Findings

Solvability of equations in whole space, half space, and bounded domains.

New embeddings for fractional parabolic Sobolev spaces.

Elementary proof techniques for embeddings.

Abstract

We consider time fractional parabolic equations in both divergence and non-divergence form when the leading coefficients $a^{ij}$ are measurable functions of $(t,x_1)$ except for $a^{11}$ which is a measurable function of either $t$ or $x_1$. We obtain the solvability in Sobolev spaces of the equations in the whole space, on a half space, or on a partially bounded domain. The proofs use a level set argument, a scaling argument, and embeddings in fractional parabolic Sobolev spaces for which we give a direct and elementary proof.