Uniform Boundedness and Convergence of Solutions to Cross-Diffusion Systems

Uniform boundedness and convergence of global solutions are proved for cross-diffusion systems in population dynamics. Gagliardo-Nirenberg type inequalities are used in the estimates of solutions in order to establish W₂¹-bounds uniform in time. In each step of estimates the contribution of the diffusion coefficients are emphasized, and it is concluded that the uniform bound remains independent of the growth of the diffusion coefficient in the system. Hence convergence of solutions are established for systems with large diffusion coefficients.