Global existence and asymptotic behavior of classical solutions to a parabolic-elliptic chemotaxis system with logistic source on ${\mathbb{R}}^N$

In the current paper, we consider the following parabolic-elliptic semilinear Keller-Segel model on ${\mathbb{R}}^N$, \begin{equation*} \begin{cases} u_{t}=\nabla\cdot (\nabla u-\chi u\nabla v)+a u -b u^2, \quad x\in$\mathbb{R}$^N,,, t>0\cr 0=(\Delta- I)v+ u, \quad x\in$\mathbb{R}$^N,,, t>0, \end{cases} \end{equation*} where $\chi >0,a\ge 0,b>0$ are constant real numbers and $N$ is a positive integer. We first prove the local existence and uniqueness of classical solutions $(u(x,t;u_0),v(x,t;u_0))$ with $u(x,0;u_0)=u_0(x)$ for various initial functions $u_0(x)$. Next, under some conditions on the constants $a,b,\chi$ and the dimension $N$, we prove the global existence and boundedness of classical solution $(u(x,t;u_0),v(x,t;u_0))$ for given initial functions $u_0(x)$. Finally, we investigate the asymptotic behavior of the global solutions with strictly positive initial functions or nonnegative compactly supported initial functions. Under some conditions on the constants $a,b,\chi$ and the dimension $N$, we show that for every strictly positive initial function $u_0(\cdot )$, $$\underset{t\to \mathrm{\infty }}{\lim }\underset{x\in {\mathbb{R}}^N}{\sup }[\mathrm{\mid }u(x,t;u_0)-\frac{a}{b}\mathrm{\mid }+\mathrm{\mid }v(x,t;u_0)-\frac{a}{b}\mathrm{\mid }]=0,$$ and that for every nonnegative initial function $u_0(\cdot )$ with non-empty and compact support $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(u_0)$, there are $0<c_{\rm low}^(u_0)\leq c_{\rm up}^(u_0)<\infty$suchthat$$\lim_{t\to\infty} \sup_{|x|\leq ct} \big[|u(x,t;u_0)-\frac{a}{b}|+|v(x,t;u_0)-\frac{a}{b}|\big]=0\quad \forall,, 0<c<c_{\rm low}^(u_0)$$and$$\lim_{t\to\infty}\sup_{|x|\geq ct} \big[u(x,t;u_0)+v(x,t;u_0)\big]=0\quad \forall,, c>c_{\rm up}^(u_0).$$