Global classical solutions, stability of constant equilibria, and spreading speeds in attraction-repulsion chemotaxis systems with logistic source on $\mathbb{R}^{N}$

We consider the following chemotaxis systems $$\begin{cases}u_t=\Delta u-\chi_1\nabla(u\nabla v_1)+\chi_2\nabla(u\nabla v_2)+u(a-bu),\ \ x\in\mathbb R^N,t>0,\\0=(\Delta-\lambda_1I)v_1+\mu_1u,\ \ x\in\mathbb R^N,t>0,\\0=(\Delta-\lambda_2I)v_2+\mu_2u,\ \ \text{in}\ x\in\mathbb R^N,\ t>0,\\u(\cdot,0)=u_0,\ \ x\in\mathbb R^N,\end{cases}$$where $\chi_i,\ \lambda_i,\ \mu_i,\ i=1,2$ and $a,\ b$ are positive constant real numbers and $N$ is a positive integer. Under some conditions on the parameters, 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 initials $u_0(x)$. Next, we show that, for every strictly positive initial \,$u_0(x)$,$$\lim_{t\to\infty}\left[\|u(\cdot,t;u_0)-\frac{a}{b}\|_{\infty}+\|\lambda_1v_1(\cdot,t;u_0)-\frac{a}{b}\mu_1\|_{\infty}+\|\lambda_2v_2(\cdot,t;u_0)-\frac{a}{b}\mu_2\|_{\infty}\right]=0.$$ Finally, we explore the spreading properties of the global solutions and prove that there are two positive numbers $0<c^*_-(\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2)<c^*_+(\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2)$ such that for every nonegative initial $u_0(x)$ with nonempty and compact support, $$\lim_{t\to\infty}\left[\sup_{|x|\leq{ct}}|u(x,t;u_0)-\frac{a}{b}|+\sup_{|x|\leq ct}|\lambda_1v_1(x,t;u_0)-\frac{a}{b}\mu_1|+\sup_{|x|\leq ct}|\lambda_2v_2(x,t;u_0)-\frac{a}{b}\mu_2|\right]=0$$whenever $0\leq c<c^*_-(\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2)$,\ and$$\lim_{t\to\infty}\left[\sup_{|x|\geq ct}|u(x,t;u_0)|+\sup_{|x|\geq ct} | v_1(x,t;u_0)|+\sup_{|x|\geq ct}|v_2(x,t;u_0)|\right]=0$$whenever $c>c^*_+(\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2)$. Furthermore we show that$$\lim_{(\chi_1,\chi_2)\to(0,0)}c^*_-(\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2)=\lim_{(\chi_1,\chi_2)\to(0,0)}c^*_+(\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2)=2\sqrt{a}.$$