Parabolic-elliptic chemotaxis model with space-time dependent logistic sources on ${\mathbb{R}}^N$. II. Existence, uniqueness, and stability of strictly positive entire solutions

This work is the second of the series of three papers devoted to the study of asymptotic dynamics in the chemotaxis system with space and time dependent logistic source,$${\mathrm{\partial }}_{t}u=\mathrm{\Delta }u-\chi \mathrm{\nabla }\cdot (u\mathrm{\nabla }v)+u(a(x,t)-ub(x,t)),0=\mathrm{\Delta }v-\lambda v+\mu u,x\in {\mathbb{R}}^N,(1)$$where $N\ge 1$ is a positive integer, $\chi ,\lambda ,\mu >0$, and the functions $a(x,t),b(x,t)$ are positive and bounded. In the first of the series, we studied the phenomena of pointwise and uniform persistence, and asymptotic spreading for solutions with compactly supported or front like initials. In the second of the series, we investigate the existence, uniqueness and stability of strictly positive entire solutions. In this direction, we prove that, if 0\le\mu\chi<b_\inf, then (1) has a strictly positive entire solution, which is time-periodic (respectively time homogeneous) when the logistic source function is time-periodic (respectively time homogeneous). Next, we show that there is positive constant ${\chi }_0$, such that for every $0\le \chi <{\chi }_0$, (1) has a unique positive entire solution which is uniform and exponentially stable with respect to strictly positive perturbations. In particular, we prove that ${\chi }_0$ can be taken to be b_\inf/2\mu when the logistic source function is either space homogeneous or the function $b(x,t)\mathrm{/}a(x,t)$ is constant. We also investigate the disturbances to Fisher-KKP dynamics caused by chemotatic effects, and prove that$$\underset{0<\chi \le {\chi }_1}{\sup }\underset{t_0\in \mathbb{R},t\ge 0}{\sup }\frac{1}{\chi }\mathrm{\parallel }{u}_{\chi }(\cdot ,t+t_0;t_0,u_0)-u_0(\cdot ,t+t_0;t_0,u_0){\mathrm{\parallel }}_{\mathrm{\infty }}<\mathrm{\infty }$$for every 0<\chi_1<b_\inf/\mu and every uniformly continuous initial function $u_0$, with $u_{0\inf }>0$, where $(u_{\chi }(x,t+t_0;t_0,u_0),v_{\chi }(x,t+t_0;t_0,u_0))$ denotes the unique classical solution of (1) with $u_{\chi }(x,t_0;t_0,u_0)=u_0(x)$, for every 0\le\chi<b_\inf.