System of Porous Medium Equations

2018/12/28 by Kim, Sunghoon, Lee, Ki-Ahm
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1812.11007

Abstract

We investigate the evolution of population density vector, \boldu=(u¹,⋯,u^k), of k-species whose diffusion is controlled by its absolute value |\boldu|. More precisely we study the properties and asymptotic large time behaviour of solution \boldu=(u¹,⋯,u^k) of degenerate parabolic system (uⁱ)_t=∇⋅(|\boldu|^m-1∇ uⁱ) for mgt;1 and i=1,⋯,k. Under some regularity assumption, we prove that the function uⁱ which describes the population density of i-th species with population M_i converges to \fracM_i|\boldM|B_|\boldM| in space with two different approaches where B_|\boldM| is the Barenblatt solution of the porous medium equation with L¹-mass |\boldM|=√(M₁²+⋯+M_k²). \indent As an application of the asymptotic behaviour, we establish a suitable harnack type inequality which makes the spatial average of uⁱ under control by the value of uⁱ at one point. We also find an 1-directional travelling wave type solutions and the properties of solutions which has travelling wave behaviour at infinity.

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