Dynamics in chemotaxis models of parabolic-elliptic type on bounded domain with time and space dependent logistic sources

This paper considers the dynamics of the following chemotaxis system $$\begin{cases} u_t=\Delta u-\chi\nabla (u\cdot \nabla v)+u\left(a_0(t,x)-a_1(t,x)u-a_2(t,x)\int_{\Omega}u\right),\quad x\in \Omega\cr 0=\Delta v+ u-v,\quad x\in \Omega \quad \cr \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=0,\quad x\in\partial\Omega, \end{cases}$$ where $\Omega \subset \mathbb{R}^n(n\geq 1)$ is a bounded domain with smooth boundary $\partial\Omega$ and $a_i(t,x)$ ($i=0,1,2$) are locally H\"older continuous in $t\in\mathbb{R}$ uniformly with respect to $x\in\bar{\Omega}$ and continuous in $x\in\bar{\Omega}$. We first prove the local existence and uniqueness of classical solutions $(u(x,t;t_0,u_0),v(x,t;t_0,u_0))$ with $u(x,t_0;t_0,u_0)=u_0(x)$ for various initial functions $u_0(x)$. Next, under some conditions on the coefficients $a_1(t,x)$, $a_2(t,x)$, $\chi$ and $n$, we prove the global existence and boundedness of classical solutions $(u(x,t;t_0,u_0),v(x,t;t_0,u_0))$ with given nonnegative initial function $u(x,t_0;t_0,u_0)=u_0(x)$. Then, under the same conditions for the global existence, we show that the system has an entire positive classical solution $(u^*(x,t),v^*(x,t))$. Moreover, if $a_i(t,x)$ $(i=0,1,2)$ are periodic in $t$ with period $T$ or are independent of $t$, then the system has a time periodic positive solution $(u^*(x,t),v^*(x,t))$ with periodic $T$ or a steady state positive solution $(u^*(x),v^*(x))$. If $a_i(t,x)$ $(i=0,1,2)$ are independent of $x$ , then the system has a spatially homogeneous entire positive solution $(u^*(t),v^*(t))$. Finally, under some further assumptions, we prove that the system has a unique entire positive solution $(u^*(x,t),v^*(x,t))$ which is globally stable . Moreover, if $a_i(t,x)$ $(i=0,1,2)$ are periodic or almost periodic in $t$, then $(u^*(x,t),v^*(x,t))$ is also periodic or almost periodic in $t$.