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

摘要

This paper considers the dynamics of the following chemotaxis system $$\begin{cases} u_t=\Delta u-\chi\nabla (u\cdot \nablav)+u\left(a_0(t,x)-a_1(t,x)u-a_2(t,x)\int_{\Omega}u\right),\quad x\in \Omega\cr0=\Delta v+ u-v,\quad x\in \Omega \quad \cr \frac{\partial u}{\partialn}=\frac{\partial v}{\partial n}=0,\quad x\in\partial\Omega, \end{cases} $$where $\Omega \subset \mathbb{R}^n(n\geq 1)$ is a bounded domain with smoothboundary $\partial\Omega$ and $a_i(t,x)$ ($i=0,1,2$) are locally H\"oldercontinuous in $t\in\mathbb{R}$ uniformly with respect to $x\in\bar{\Omega}$ andcontinuous in $x\in\bar{\Omega}$. We first prove the local existence anduniqueness 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, undersome conditions on the coefficients $a_1(t,x)$, $a_2(t,x)$, $\chi$ and $n$, weprove 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 globalexistence, 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 periodicpositive solution $(u^*(x,t),v^*(x,t))$ with periodic $T$ or a steady statepositive solution $(u^*(x),v^*(x))$. If $a_i(t,x)$ $(i=0,1,2)$ are independentof $x$ , then the system has a spatially homogeneous entire positive solution$(u^*(t),v^*(t))$. Finally, under some further assumptions, we prove that thesystem has a unique entire positive solution $(u^*(x,t),v^*(x,t))$ which isglobally stable . Moreover, if $a_i(t,x)$ $(i=0,1,2)$ are periodic or almostperiodic in $t$, then $(u^*(x,t),v^*(x,t))$ is also periodic or almost periodicin $t$.