Quasilinear divergence form parabolic equations in Reifenberg flat domains

We derive weak solvability and higher integrability of the spatial gradient of solutions to Cauchy-Dirichlet problem for divergence form quasi-linear parabolic equations {u(t) - div (a(ij)(x, t, u)D(j)u + a(i)(x, t, u)) = b(x, t, u, Du) in Q, u = 0 on partial derivative(p)Q, where Q is a cylinder in R(n) x (0, T) with Reifenberg flat base Omega. The principal coefficients a(ij)(x, t, u) of the uniformly parabolic operator are supposed to have small BMO norms with respect to (x, t) while the nonlinear terms a(i)(x, t, u) and b(x, t, u, Du) support controlled growth conditions.