Equilibration in a Fully Parabolic Two-Species Chemotaxis System with Competitive Kinetics

The fully parabolic two-species chemotaxis system (*) [GRAPHICS] is considered in a bounded domain Omega subset of R-n with smooth boundary. It is shown that if n <= 2 and all parameters in (*) are merely positive, then for all appropriately regular nonnegative initial data u(0), v(0), and w(0), the corresponding Neumann initial-boundary value problem possesses a unique global bounded solution. Moreover, by means of the construction of suitable energy functionals it is proved that, whenever n >= 1, the following hold: If a(1) < 1 and a(2) < 1 and both mu(1) and mu(2) are sufficiently large, then any global bounded solution emanating from adequately regular initial data fulfilling u(0) not equivalent to 0 not equivalent to v(0) satisfies (u, v, w)(., t) -> (u(star,) v(star), w(star)) uniformly in Omega as t -> infinity, where (u(star), v(star), w(star)) denotes the unique positive spatially homogeneous equilibrium of (star). If a(1) >= 1 and a(2) < 1, and mu(2) is large enough, then all global bounded solutions with reasonably smooth initial data satisfying v(0) not equivalent to 0 have the property that (u, v, w) (., t) -> (0,1,beta/gamma) uniformly in Omega as t -> infinity. The respective rates of convergence are shown to be at least exponential when a(1) not equal 1, and algebraic if a(1) = 1.