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Persistence and convergence in parabolic-parabolic chemotaxis system with logistic source on $ \mathbb{R}^{N} $

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  • In the current paper, we consider the following parabolic-parabolic chemotaxis system with logistic source on$ \mathbb{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>\frac{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>\frac{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)>\frac{N}{4} $such that if$ b>K \chi\mu $and$ \lambda\geq \frac{a}{2} $, then for every strictly positive initial function$ u_0(\cdot) $, it holds that

    $  \lim\limits_{t\to\infty}\big[\|u(x,t;u_0, v_0)-\frac{a}{b}\|_{\infty}+\|v(x,t;u_0, v_0)-\frac{\mu}{\lambda}\frac{a}{b}\|_{\infty}\big] = 0.  $

    Mathematics Subject Classification: 35A01, 35B35, 35B40, 35Q92, 92C17.

    Citation:

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