$L_p$-estimates for parabolic equations in divergence form with a half-time derivative

TL;DR

This paper proves the unique solvability of linear parabolic equations with a half-time derivative in Sobolev spaces, accommodating irregular coefficients and expanding the class of solvable evolution equations.

Contribution

It introduces a novel framework for solving parabolic equations with half-time derivatives and irregular coefficients, extending existing theories.

Findings

Established unique solvability in Sobolev spaces.

Handled equations with merely measurable coefficients.

Included half-time derivatives in the analysis.

Abstract

We establish the unique solvability of solutions in Sobolev spaces to linear parabolic equations in a more general form than those in the literature. A distinguishing feature of our equations is the inclusion of a half-order time derivative term on their right-hand side. We anticipate that such equations will prove useful in various problems involving time evolution terms. Notably, the coefficients of the equations exhibit significant irregularity, being merely measurable with respect to the temporal variable or one spatial variable.