The effects of cross-diffusion and logistic source on the boundedness of solutions to a pursuit-evasion model

We study the following quasilinear pursuit-evasion model: [GRAPHICS] in a smooth and bounded domain SZ subset of Rn(n >= 1), where a, b, chi, xi, lambda 1, lambda 2, mu 1, mu 2 > 0, alpha, beta E R, and r1, r2 > 1. When r1 > max{1, 1 + alpha}, r2 > max{1, 1 +beta}, it has been proved that if min{(r1 - 1)(r2 - beta - 1), (r1 - alpha - 1)(r2 -beta - 1)} > (n-2)+ n , then for some suitable nonnegative initial data u0 and v0, the system admits a unique globally classical solution which is bounded in SZ x (0, infinity).