Existence theorem for a partially parabolic cross-diffusion system

We study an initial boundary value problem for a cross-diffusion system in population dynamics. The mathematical challenge is due to the fact that the determinant of the coefficient matrix of the system changes signs. As a result, the system is only partially parabolic. We design an approximation scheme. The sequence of approximate solutions generated by our scheme converges and its limit satisfies the original system in the parabolic region. It remains open if one can construct a vector-valued function that satisfies the system in both the parabolic region and the anti-parabolic one.