Local continuity and asymptotic behaviour of degenerate parabolic systems. (March 2020) - Kim, Sunghoon
Abstract: We study the local Hölder continuity and the asymptotic behaviour of solution, u = ( u 1, …, u k ), of the degenerate system u t i = ∇ ⋅ m U m − 1 ∇ u i for m > 1 and i = 1, …, k which describes the population densities of k -species whose diffusions are determined by their total population density U = u 1 + ⋯ + u k . 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 i th species with the population M i converges in C s ∞ to the function M i M B M ( x, t ) as t → ∞ where B M is the Barenblatt profile of the standard porous medium equation with L 1 mass M = M 1 + ⋯ + M k . As a consequence of asymptotic behaviour, it is shown that each density becomes a concave function after a finite time.