Persistence and convergence in parabolic-parabolic chemotaxis system with logistic source on RN
Abstract
In the current paper, we consider the following parabolic-parabolic chemotaxis system with logistic source on RN, equation cases ut= u-∇· ( u∇ v) + u(a-bu), x∈RN\,\,\, t>0 vt= v -λ v+μ u, x∈ RN\,\,\, t>0 cases(1) equation where , \ a,\ b,\ λ,\ μ 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>Nμ4, the global existence of a unique classical solution (u(x,t;u0, v0),v(x,t;u0, v0)) of (1) with u(x,0;u0, v0)=u0(x) and v(x,0;u0, v0)=v0(x) for every nonnegative, bounded, and uniformly continuous function u0(x), and every nonnegative, bounded, uniformly continuous, and differentiable function v0(x). Next, under the same assumption b>Nμ4, we show that persistence phenomena occurs, that is, any globally defined bounded positive classical solution with strictly positive initial function u0 is bounded below by a positive constant independent of (u0, v0) when time is large. Finally, we discuss the asymptotic behavior of the global classical solution with strictly positive initial function u0. We show that there is K=K(a,λ,N)>N4 such that if b>K μ and λ≥ a2, then for every strictly positive initial function u0(·), it holds that t∞[\|u(x,t;u0, v0)-ab\|∞+\|v(x,t;u0, v0)-μλab\|∞]=0.