Regularity in parabolic systems with continuous coefficients

Under the only assumption of continuous coefficients, we prove a partial Hölder continuity result for solutions to parabolic systems with polynomial growth. A key component throughout the argument is the use of DiBenedetto’s intrinsic geometry (Degenerate Parabolic Equations. Universitext. Springer, New York, 1993) to accommodate the inhomogeneity in the system. A main technical point is that, although we are proving the Hölder continuity of solutions, we employ the intrinsic geometry using cylinder stretched according to the size of the (spatial) gradient, in an iteration scheme that does not necessarily imply the boundedness of the gradient itself