L-estimates for time fractional parabolic equations in divergence form with measurable coefficients

Abstract

In this paper, we establish Lp 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 with respect to the other variables. The corresponding results for equations on a half space are also derived.2010 Mathematics Subject Classification. 35R11, 26A33, 35R05.

On the trace embedding and its applications to evolution equations

“…Remark 6.4. A special class of Volterra type equations in the case of coefficients which are (t, x)-dependent is treated in [16,17,18] assuming only measurability in t and very weak conditions in space. They prove maximal regularity results, and their approach is completely different than the one considered here.…”

Recent Progress in the $L_p$ Theory for Elliptic and Parabolic Equations with Discontinuous Coefficients

“…In this direction, we also refer the reader to [41] for a corresponding result for divergence form equations with non-local time derivatives with leading coefficients measurable in a space variable and VMO with respect to other variables, as well as [54] for an extension of the result in [65] to weighted mixed-norm Lebesgue spaces for the fractional heat equation −∂ α t u + ∆u = f . Given the results in [39] and [54,65], it is natural to ask whether the result in [40] can be extended to the mixed-norm spaces and whether it is possible to also include weights.…”

The Riemann-Liouville fractional integral in Bochner-Lebesgue spaces I

“…After its conception as an area of expertise, the fractional calculus have gone through a huge paradigm shift, going from being a forgotten subject to a trending one; today we may cite several papers that have been published in important mathematical journals: see [2,6,7,8,9,10,14,18,19,24,27] as a sample of this literature.…”