On solvability of parabolic equations with singular coefficients in odd mixed-norm Morrey-Sobolev spaces

TL;DR

This paper establishes existence and uniqueness results for second-order parabolic equations with singular coefficients in novel mixed-norm Morrey-Sobolev spaces, accommodating rough coefficients and singularities.

Contribution

It introduces and analyzes odd mixed-norm Morrey-Sobolev spaces for parabolic equations with singular coefficients, proving solvability under minimal regularity assumptions.

Findings

Proved existence and uniqueness of solutions in the new function spaces.

Extended solvability results to equations with rough coefficients and singularities.

Demonstrated the applicability of odd mixed-norm Morrey-Sobolev spaces to parabolic PDEs.

Abstract

We prove an existence and uniqueness theorem for second-order parabolic equations in the whole space with constant zeroth-order coefficient in mixed-norm Morrey-Sobolev spaces. The main coefficient $a$ is assumed to be measurable in $t$ and BMO in $x$ and the first-order coefficients $b$ are in an appropriate mixed-norm Morrey classes (thus admitting rather rough singularities). The mixed-norm Morrey-Sobolev spaces are ``odd'' in the sense that the interior integration in the formula defining the norm is performed with respect to $t$ and not to $x$ as is customary.