Time Fractional Parabolic Equations with Measurable Coefficients and Embeddings for Fractional Parabolic Sobolev Spaces

Abstract

Abstract We consider time fractional parabolic equations in 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, and 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.