Local Continuity and Asymptotic Behaviour of Degenerate Parabolic Systems

2018/03/17 by Kim, Sunghoon, Lee, Ki-Ahm
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1803.06465

Abstract

We study the local Hölder continuity and the asymptotic behaviour of solution, u=(u¹,⋯, u^k), of the degenerate system uⁱ_t=∇⋅(m U^m-1∇ uⁱ) for mgt;1 and i=1,⋯,k which describes the populations density of k-species whose diffusion is determined by their total population density U=u¹+⋯+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 function of i-th species with the population M_i converges in C_s^∞ to (M_i)/(M)B_M(x,t) as t→ ∞ where B_M is the Barenblatt profile of the standard porous medium equation with L¹ mass M=M₁+⋯+M_k. As a consequence of asymptotic behaviour, it is shown that each density function becomes a concave function after a finite time.

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