PERSISTENCE AND CONVERGENCE IN PARABOLIC-PARABOLIC CHEMOTAXIS SYSTEM WITH LOGISTIC SOURCE ON RN

In the current paper, we consider the following parabolic-parabolic chemotaxis system with logistic source on R-N, {u(t) = Delta u - chi del center dot(u del v) + u(a-bu), x is an element of R-N, (1) v(t) = Delta v - lambda v + mu u, x is an element of R-N, where chi, a, b, lambda, mu are positive constants and N is a positive integer. We investigate the persistence and convergence in (1). To this end, we first prove, under the assumption b > N chi mu/4, the global existence of a unique classical solution (u(x, t; u(0), v(0)), v(x, t; u(0), v(0))) of (1) with u(x, 0; u(0), v(0)) = u(0)(x) and v (x, 0; u(0), v(0)) = v(0)(x) for every nonnegative, bounded, and uniformly continuous function u(0)(x), and every nonnegative, bounded, uniformly continuous, and differentiable function v(0)(x). Next, under the same assumption b > N chi mu/4, we show that persistence phenomena occurs, that is, any globally defined bounded positive classical solution with strictly positive initial function u(0) is bounded below by a positive constant independent of (u(0), v(0)) when time is large. Finally, we discuss the asymptotic behavior of the global classical solution with strictly positive initial function u(0). We show that there is K = K (a, lambda, N) > N/4 such that if b > K chi mu and lambda >= a/2, then for every strictly positive initial function u(0)(center dot), it holds that lim(t ->infinity) [parallel to u(x, t; u(0), v(0)) - a/b parallel to(infinity) + parallel to v(x, t; u(0), v(0)) - mu/lambda a/b parallel to(infinity)] = 0.