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On the steady-states of a two-species non-local cross-diffusion model

Rodríguez Nancy, Hu Yi

Journal of Applied Analysis

2020

Vol. 26, nr 1

1--19

We investigate the existence and properties of steady-state solutions to a degenerate, non-local system of partial differential equations that describe two-species segregation in homogeneous and heterogeneous environments. This is accomplished via the analysis of the existence and non-existence of global minimizers to the corresponding free energy functional. We prove that in the spatially homogeneous case global minimizers exist if and only if the mass of the potential governing the intra-species attraction is sufficiently large and the support of the potential governing the interspecies repulsion is bounded. Moreover, when they exist they are such that the two species have disjoint support, leading to complete segregation. For the heterogeneous environment we show that if a sub-additivity condition is satisfied then global minimizers exists. We provide an example of an environment that leads to the sub-additivity condition being satisfied. Finally, we explore the bounded domain case with periodic conditions through the use of numerical simulations.

On boundary value problem for a nonlocal elliptic equation

Fijałkowski P., Przeradzki B.

Journal of Applied Analysis

2003

Vol. 9, nr 2

201-209

The existence of a positive radial solution to the Dirichlet boundary value problem for the second order elliptic equation [wzór], where U = B(0, R) \ ‾B(0, ρ), with weak assumptions on the nonlinear term f, is proved. The method based on the Krasnosel'skii Fixed Point Theorem enables to find many solutions to the problem. Solutions for the same problem but with U = B(0, R) and with nonlinear term f depending explicitely on |x| are found as well.