Regularity of the solutions to quasi-linear parabolic systems with the singular coefficients

This article establishes the regularity properties of solutions to the parabolic quasilinear parabolic systems in the divergent form ∂/∂t⃗u, −d/dxi⃗ai (x, t, ⃗u, ∇⃗u) +⃗b (x, t, ⃗u, ∇⃗u) = 0, under rather general conditions on its coefficients. To prove solvability, we apply the Leray-Schauder theory and method of apriori estimations.