# Regularity gradient estimates for weak solutions of singular quasi-linear parabolic equations

## TL;DR

This paper establishes new weighted Sobolev regularity estimates for weak solutions of singular quasi-linear parabolic equations with discontinuous coefficients, advancing understanding even for linear cases.

## Contribution

It provides the first global and interior weighted regularity estimates for solutions with singular, discontinuous coefficients depending on the solution itself.

## Key findings

- Established weighted W^{1,p} regularity estimates for weak solutions.
- Results are new even for linear equations with singular coefficients.
- Addresses regularity in the presence of discontinuous, solution-dependent coefficients.

## Abstract

This paper studies the Sobolev regularity estimates of weak solutions of a class of singular quasi-linear elliptic problems of the form $u_t - \mbox{div}[\mathbb{A}(x,t,u,\nabla u)]= \mbox{div}[{\mathbf F}]$ with homogeneous Dirichlet boundary conditions over bounded spatial domains. Our main focus is on the case that the vector coefficients $\mathbb{A}$ are discontinuous and singular in $(x,t)$-variables, and dependent on the solution $u$. Global and interior weighted $W^{1,p}(\Omega, \omega)$-regularity estimates are established for weak solutions of these equations, where $\omega$ is a weight function in some Muckenhoupt class of weights. The results obtained are even new for linear equations, and for $\omega =1$, because of the singularity of the coefficients in $(x,t)$-variables