Local continuity and asymptotic behaviour of degenerate parabolic systems

We study the local Hölder continuity and the asymptotic behaviour of solution, u=(u1,…,uk), of the degenerate system uti=∇⋅mUm−1∇uifor m>1 and i=1,…,kwhich describes the population densities of k-species whose diffusions are determined by their total population density U=u1+⋯+uk. For the local Hölder continuity, we adopt the intrinsic scaling and iteration arguments of DeGiorgi, Moser, and DiBenedetto. Under some regularity conditions, we also prove that the population density of ith species with the population Mi converges in Cs∞ to the function MiMBM(x,t) as t→∞ where BM is the Barenblatt profile of the standard porous medium equation with L1 mass M=M1+⋯+Mk. As a consequence of asymptotic behaviour, it is shown that each density becomes a concave function after a finite time.