The regularity of weak solutions for certain n-dimensional strongly coupled parabolic systems

This paper is concerned with the -dimensional strongly coupled parabolic systems with triangular form in the cylinder . We investigate and Hölder regularity of the derivatives of weak solutions for the systems in the following two cases: one is that the boundedness of and has not been shown in existence result of solutions; the other is that the boundedness of or has been shown in existence result of solutions. By using difference ratios and Steklov averages methods and various estimates, we prove that if is a weak solution of the system, then for any and , belong to and under certain conditions, and belong to under stronger assumptions. Applications of these results are given to two ecological models with cross-diffusion.