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Journal of Dynamics and Differential Equations

, Volume 31, Issue 3, pp 1301–1325

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

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## Abstract

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}

## Keywords

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

## Mathematics Subject Classification

35B35 35B40 35K57 35Q92 92C17

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## Copyright information

© Springer Science+Business Media, LLC 2017

## Authors and Affiliations

• Rachidi B. Salako
• 1
• Wenxian Shen
• 1
1. 1.Department of Mathematics and StatisticsAuburn UniversityAuburnUSA

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