System of porous medium equations

We investigate the evolution of population density vector, u = (u(1),..., u(k)), of k-species whose diffusion is controlled by its absolute value vertical bar u vertical bar. More precisely, we study the properties and asymptotic large time behaviour of solution (u(1),..., u(k)) of degenerate parabolic system (u(i))(t) = del . (vertical bar u vertical bar(m-1) del u(i)) for m > 1 and i = 1,..., k. Under some regularity assumptions, we prove that the component u(i) which describes the population density of i-th species with population M-i converges to Mi/vertical bar M vertical bar B-vertical bar M vertical bar in space with two different approaches where B-vertical bar M vertical bar is the Barenblatt solution of the standard porous medium equation with L-1 mass vertical bar M vertical bar = root M-1(2) +....+ M-k(2). As an application of the asymptotic behaviour, we establish a suitable Harnack type inequality which makes the spatial average of u(i) under control by the value of u(i) at one point. We also find 1-directional travelling wave type solutions and the properties of solutions which has travelling wave behaviour at infinity. (C) 2020 Elsevier Inc. All rights reserved.

Automated summary

The study investigates the evolution of population density vector of k-species controlled by the absolute value |u|, and the properties and asymptotic behavior of the degenerate parabolic system. By proving the convergence of population densities and establishing a suitable Harnack type inequality, the spatial behavior of population densities is revealed in different approaches. The study also identifies 1-directional travelling wave solutions and properties of solutions exhibiting travelling wave behavior at infinity.