Uniform boundedness and convergence of solutions to the systems with a single nonzero cross-diffusion

Uniform boundedness and convergence of global solutions are proved for quasilinear parabolic systems with a single nonzero cross-diffusion in population dynamics. Gagliardo–Nirenberg type inequalities are used in the estimates of solutions in order to establish W1 2 -bounds uniform in time. By using the uniform bound, convergence of solutions are established for systems with large diffusion coefficients in the weak competition case.  2003 Elsevier Science (USA). All rights reserved.