On Global Existence of Solutions to a Cross-diffusion System

the Laplacian, ∂/∂ν denotes the directional derivative along the outward normal on ∂Ω, ai, bi, ci, di (i = 1, 2) are given positive constants and α, γ, δ, β are nonnegative constants. In the system (1.1) u and v are non-negative functions which represent population densities of two competing species, d1 and d2 are respectively their diffusion rates. Parameters a1 and a2 are intrinsic growth rates, b1 and c2 are coefficients for intra-specific competitions, b2 and c1 are coefficients for inter-specific competitions. Parameters γ and δ are usually called self-diffusion rates, and α and β are called cross-diffusion rates. The homogeneous Neumann boundary condition means there is no migration crossing the boundary ∂Ω. When α = γ = δ = β = 0, (1.1) reduces to the well-known Lotka-Volterra competition-diffusion system. Mathematically, the problem (1.1) has received a lot of attention. Local existence (in time) of solutions to (1.1) was established by Amann in a series of important papers [1], [2], [3]. His results can be summarized as follows.