Journal of Dynamics and Differential Equations

, Volume 31, Issue 3, pp 1301–1325 | Cite as

Global Classical Solutions, Stability of Constant Equilibria, and Spreading Speeds in Attraction–Repulsion Chemotaxis Systems with Logistic Source on \(\mathbb {R}^{N}\)

  • Rachidi B. Salako
  • Wenxian ShenEmail author


In this paper, we consider the following chemotaxis systems of parabolic–elliptic–elliptic type on \(\mathbb {R}^{N}\),
$$\begin{aligned} {\left\{ \begin{array}{ll} u_{t}=\Delta u -\chi _1\nabla ( u\nabla v_1)+\chi _2 \nabla (u\nabla v_2 )+ u(a -b u), \qquad \ x\in {\mathbb R}^N,\ t>0, \\ 0=(\Delta - \lambda _1 I)v_1+ \mu _1 u, \qquad \ x\in {\mathbb R}^N,\ t>0, \\ 0=(\Delta - \lambda _2 I)v_2+ \mu _2 u, \qquad \ x\in {\mathbb R}^N,\ t>0, \\ u(\cdot ,0)=u_{0}, \qquad x\in \mathbb {R}^N , \end{array}\right. } \end{aligned}$$
where \( \chi _{i}\ge 0,\ \lambda _i> 0,\ \mu _i>0\) (\(i=1,2\)) and \(\ a> 0,\ b> 0\) are constant real numbers, and N is a positive integer. First, under some conditions on the parameters \(\chi _i,\mu _i,\lambda _i, a, b\) and N, we prove the global existence and boundedness of classical solutions \((u(x,t;u_0),v_1(x,t;u_0),v_2(x,t;u_0))\) for nonnegative, bounded, and uniformly continuous initial functions \(u_0(x)\). Next, we explore the asymptotic stability of the constant equilibrium \((\frac{a}{b},\frac{\mu _1}{\lambda _1}\frac{a}{b},\frac{\mu _2}{\lambda _2}\frac{a}{b})\) and prove under some further assumption on the parameters that, for every strictly positive initial \(u_0(x)\),
$$\begin{aligned} \lim _{t\rightarrow \infty }\left[ \Vert u(\cdot ,t;u_0)-\frac{a}{b}\Vert _{\infty }+\Vert \lambda _{1}v_{1}(\cdot ,t;u_0)-\frac{a}{b}\mu _1\Vert _{\infty }+\Vert \lambda _{2}v_{2}(\cdot ,t;u_0)-\frac{a}{b}\mu _2\Vert _{\infty }\right] =0. \end{aligned}$$
Finally, we investigate the spreading properties of the global solutions with compactly supported initial functions. We show that under some conditions on the parameters, there are two positive numbers \(0<c^*_-(\chi _1,\mu _1,\lambda _1,\chi _2,\mu _2,\lambda _2)\le c^*_+(\chi _1,\mu _1,\lambda _1,\chi _2,\mu _2,\lambda _2)\) such that for every nonnegative initial function \(u_0(x)\) with nonempty and compact support, we have
$$\begin{aligned} \lim _{t\rightarrow \infty }\left[ \sup _{|x|\le ct}|u(x,t;u_0)-\frac{a}{b}| {+} \sup _{|x|\le ct}|\lambda _1 v_1(x,t;u_0)-\frac{a}{b}\mu _1| {+} \sup _{|x|\le ct}|\lambda _2 v_2(x,t;u_0)-\frac{a}{b}\mu _2|\right] =0 \end{aligned}$$
whenever \(0\le c< c^*_-(\chi _1,\mu _1,\lambda _1,\chi _2,\mu _2,\lambda _2)\), and
$$\begin{aligned} \lim _{t\rightarrow \infty }\left[ \sup _{|x|\ge ct}|u(x,t;u_0)| + \sup _{|x|\ge ct}| v_1(x,t;u_0)| + \sup _{|x|\ge ct}|v_2(x,t;u_0)|\right] =0 \end{aligned}$$
whenever \(c>c^*_+(\chi _1,\mu _1,\lambda _1,\chi _2,\mu _2,\lambda _2)\). Furthermore we show that
$$\begin{aligned}\lim _{(\chi _1,\chi _2)\rightarrow (0,0)}c^*_-(\chi _1,\mu _1,\lambda _1,\chi _2,\mu _2,\lambda _2)=\lim _{(\chi _1,\chi _2)\rightarrow (0,0)}c^*_+(\chi _1,\mu _1,\lambda _1,\chi _2,\mu _2,\lambda _2)=2\sqrt{a}. \end{aligned}$$


Parabolic–elliptic chemotaxis system Logistic source Classical solution Local existence Global existence Asymptotic stability Spreading speeds 

Mathematics Subject Classification

35B35 35B40 35K57 35Q92 92C17 



The authors would like to thank the referee for valuable comments and suggestions which improved the presentation of this paper considerably.


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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsAuburn UniversityAuburnUSA

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