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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.
We establish the $L_p$-solvability for time fractional parabolic equations when coefficients are merely measurable in the time variable. In the spatial variables, the leading coefficients locally have small mean oscillations. Our results extend a recent result in [6] to a large extent.
We consider both divergence and non-divergence parabolic equations on a half space in weighted Sobolev spaces. All the leading coefficients are assumed to be only measurable in the time and one spatial variable except one coefficient, which is assume
In this paper, we establish $L_p$ estimates and solvability for time fractional divergence form parabolic equations in the whole space when leading coefficients are merely measurable in one spatial variable and locally have small mean oscillations wi
We obtain $L_p$ estimates for fractional parabolic equations with space-time non-local operators $$ partial_t^alpha u - Lu= f quad mathrm{in} quad (0,T) times mathbb{R}^d,$$ where $partial_t^alpha u$ is the Caputo fractional derivative of order $alph
In this paper, we establish several Liouville type theorems for entire solutions to fractional parabolic equations. We first obtain the key ingredients needed in the proof of Liouville theorems, such as narrow region principles and maximum principles