Global estimates for quasilinear parabolic equations on Reifenberg flat domains and its applications to Riccati type parabolic equations with distributional data

TL;DR

This paper establishes global weighted Lorentz estimates for solutions to quasilinear parabolic equations on Reifenberg flat domains, leading to existence results for Riccati type equations with distributional data.

Contribution

It provides the first global weighted Lorentz and Lorentz-Morrey estimates for gradients of solutions under minimal boundary regularity, enabling new existence results for Riccati equations with measure data.

Findings

Proved global weighted Lorentz estimates for solutions.

Established existence of solutions to Riccati equations with measure data.

Extended analysis to Reifenberg flat domains with minimal boundary regularity.

Abstract

In this paper, we prove global weighted Lorentz and Lorentz-Morrey estimates for gradients of solutions to the quasilinear parabolic equations: $$u_t-\operatorname{div}(A(x,t,\nabla u))=\operatorname{div}(F),$$ in a bounded domain $\Omega\times (0,T)\subset\mathbb{R}^{N+1}$, under minimal regularity assumptions on the boundary of domain and on nonlinearity $A$. Then results yields existence of a solution to the Riccati type parabolic equations: $$u_t-\operatorname{div}(A(x,t,\nabla u))=|\nabla u|^q+\operatorname{div}(F)+\mu,$$ where $q>1$ and $\mu$ is a bounded Radon measure.