Local continuity and asymptotic behaviour of degenerate parabolic systems. (March 2020)

• Record Type: Journal Article
• Authors: Kim, Sunghoon; Lee, Ki-Ahm
• Is Part Of: Nonlinear analysis. Volume 192(2020)
• Publication Date: 2020-03
• Subjects: Local continuity -- Asymptotic behaviour -- Degenerate equation -- Eventual concavity; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2019.111702
• Languages: English
• ISSNs: 0362-546X

Abstract

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.