Parabolic-elliptic chemotaxis model with space-time dependent logistic sources on $\mathbb{R}^N$. I. Persistence and asymptotic spreading

The current series of three papers is concerned with the asymptotic dynamics in the following chemotaxis model $$\partial_tu=\Delta u-\chi\nabla(u\nabla v)+u(a(x,t)-ub(x,t))\ ,\ 0=\Delta v-\lambda v+\mu u \ \ (1)$$where $\chi, \lambda, \mu$ are positive constants, $a(x,t)$ and $b(x,t)$ are positive and bounded. In the first of the series, we investigate the persistence and asymptotic spreading. Under some explicit condition on the parameters, we show that (1) has a unique nonnegative time global classical solution $(u(x,t;t_0,u_0),v(x,t;t_0,u_0))$ with $u(x,t_0;t_0,u_0)=u_0(x)$ for every $t_0\in R$ and every $u_0\in C^{b}_{\rm unif}(R^N)$, $u_0\geq 0$. Next we show the pointwise persistence phenomena in the sense that, for any solution $(u(x,t;t_0,u_0),v(x,t;t_0,u_0))$ of (1) with strictly positive initial function $u_0$, then$$0<\inf_{t_0\in R, t\geq 0}u(x,t+t_0;t_0,u_0)\le\sup_{t_0\in R, t\geq 0} u(x,t+t_0;t_0,u_0)<\infty$$and show the uniform persistence phenomena in the sense that there are $0<m<M$ such that for any strictly positive initial function $u_0$, there is $T(u_0)>0$ such that$$m\le u(x,t+t_0;t_0,u_0)\le M\ \forall\,t\ge T(u_0),\ x\in R^N.$$We then discuss the spreading properties of solutions to (1) with compactly supported initial and prove that there are positive constants $0<c_{-}^{*}\le c_{+}^{*}<\infty$ such that for every $t_0\in R$ and every $u_0\in C^b_{\rm unif}(R^N), u_0\ge 0$ with nonempty compact support, we have that$$\lim_{t\to\infty}\sup_{|x|\ge ct}u(x,t+t_0;t_0,u_0)=0,\ \forall c>c_+^*,$$and$$\liminf_{t\to\infty}\sup_{|x|\le ct}u(x,t+t_0;t_0,u_0)>0, \ \forall 0<c<c_-^*.$$We also discuss the spreading properties of solutions to (1) with front-like initial functions. In the second and third of the series, we will study the existence, uniqueness, and stability of strictly positive entire solutions and the existence of transition fronts, respectively.