  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 freeboundary 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.
