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

Abstract

This paper studies the Sobolev regularity for weak solutions of a class of singular quasi-linear parabolic problems of the form ut−div[A(x,t,u,∇u)]=div[F] with homogeneous Dirichlet boundary conditions over bounded spatial domains. Our main focus is on the case that the vector coefficients A are discontinuous and singular in (x,t)-variables, and dependent on the solution u. Global and interior weighted W1,p(ΩT,ω)-regularity estimates are established for weak solutions of these equations, where ω is a weight function in some Muckenhoupt class of weights. The results obtained are even new for linear equations, and for ω=1, because of the singularity of the coefficients in (x,t)-variables.