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

Abstract

<p style='text-indent:20px;'>In the current paper, we consider the following parabolic-parabolic chemotaxis system with logistic source on <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^{N} $\end{document}</tex-math></inline-formula>,</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} \begin{cases} u_{t} = \Delta u - \chi\nabla\cdot ( u\nabla v) + u(a-bu),\quad x\in{{\mathbb R}}^N,\\ {v_t} = \Delta v -\lambda v+\mu u,\quad x\in{{\mathbb R}}^N,\,\,\, \end{cases} \;\;\;\;\;\;\;\;\left( 1 \right)\end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M3">\begin{document}$ \chi, \ a,\ b,\ \lambda,\ \mu $\end{document}</tex-math></inline-formula> are positive constants and <inline-formula><tex-math id="M4">\begin{document}$ N $\end{document}</tex-math></inline-formula> is a positive integer. We investigate the persistence and convergence in (1). To this end, we first prove, under the assumption <inline-formula><tex-math id="M5">\begin{document}$ b>\frac{N\chi\mu}{4} $\end{document}</tex-math></inline-formula>, the global existence of a unique classical solution <inline-formula><tex-math id="M6">\begin{document}$ (u(x,t;u_0, v_0),v(x,t;u_0, v_0)) $\end{document}</tex-math></inline-formula> of (1) with <inline-formula><tex-math id="M7">\begin{document}$ u(x,0;u_0, v_0) = u_0(x) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ v(x,0;u_0, v_0) = v_0(x) $\end{document}</tex-math></inline-formula> for every nonnegative, bounded, and uniformly continuous function <inline-formula><tex-math id="M9">\begin{document}$ u_0(x) $\end{document}</tex-math></inline-formula>, and every nonnegative, bounded, uniformly continuous, and differentiable function <inline-formula><tex-math id="M10">\begin{document}$ v_0(x) $\end{document}</tex-math></inline-formula>. Next, under the same assumption <inline-formula><tex-math id="M11">\begin{document}$ b>\frac{N\chi\mu}{4} $\end{document}</tex-math></inline-formula>, we show that persistence phenomena occurs, that is, any globally defined bounded positive classical solution with strictly positive initial function <inline-formula><tex-math id="M12">\begin{document}$ u_0 $\end{document}</tex-math></inline-formula> is bounded below by a positive constant independent of <inline-formula><tex-math id="M13">\begin{document}$ (u_0, v_0) $\end{document}</tex-math></inline-formula> when time is large. Finally, we discuss the asymptotic behavior of the global classical solution with strictly positive initial function <inline-formula><tex-math id="M14">\begin{document}$ u_0 $\end{document}</tex-math></inline-formula>. We show that there is <inline-formula><tex-math id="M15">\begin{document}$ K = K(a,\lambda,N)>\frac{N}{4} $\end{document}</tex-math></inline-formula> such that if <inline-formula><tex-math id="M16">\begin{document}$ b>K \chi\mu $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M17">\begin{document}$ \lambda\geq \frac{a}{2} $\end{document}</tex-math></inline-formula>, then for every strictly positive initial function <inline-formula><tex-math id="M18">\begin{document}$ u_0(\cdot) $\end{document}</tex-math></inline-formula>, it holds that</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \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. $\end{document} </tex-math></disp-formula></p>

Spreading speeds of a parabolic-parabolic chemotaxis model with logistic source on $ \mathbb{R}^{N} $

“…The following proposition states the existence and uniqueness of global classical solutions of (2) with non-negative initial function. It has been proved in [41,Theorem 1.2].…”

“…Very recently, the authors of current paper studied in [41] the global existence, persistence, and convergence of positive solutions of (2) with a, b > 0. It is proved in [41] that if b > N µχ 4 , (2) has a unique bounded global classical solution for every nonnegative, bounded, and uniformly continuous function u 0 (x), and every nonnegative, bounded, uniformly continuous, and differentiable function v 0 (x). Hence finite-time blow-up phenomena in (2) can also be suppressed to some extent by the logistic source.…”