On long-time dynamics for competition-diffusion systems with inhomogeneous Dirichlet boundary conditions

Abstract

We consider a two-component competition-diffusion system with equal
diffusion coefficients and inhomogeneous Dirichlet boundary conditions. When
the interspecific competition parameter tends to infinity, the system
solution converges to that of a free-boundary problem. If all stationary
solutions of this limit problem are non-degenerate and if a certain linear
combination of the boundary data does not identically vanish, then for
sufficiently large interspecific competition, all non-negative solutions of
the competition-diffusion system converge to stationary states as time tends
to infinity. Such dynamics are much simpler than those found for the
corresponding system with either homogeneous Neumann or homogeneous Dirichlet
boundary conditions.