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

DOI: http://dx.doi.org/10.12775/TMNA.2007.017

#### 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.

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.

#### Keywords

Competition-diffusion system; boundary-value problem; singular limit; long-time behaviour; spatial segregation

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