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

This paper studies the Sobolev regularity for weak solutions of a class of singular quasi-linear parabolic problems of the form u(t) - div[A(x, t,u,del 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 W-1,W-p (Omega(T),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. (C) 2017 Elsevier Inc. All rights reserved.