Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $t$.

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The categories of this paper are: gr-qc
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The categories of this paper are: hep-th
37
The categories of this paper are: supr-con
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The categories of this paper are: acc-phys
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The categories of this paper are: astro-ph
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The categories of this paper are: cs
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The categories of this paper are: chao-dyn
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The categories of this paper are: q-bio
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The categories of this paper are: hep-ph
36
The categories of this paper are: cmp-lg
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The categories of this paper are: alg-geom
31
The categories of this paper are: ao-sci
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The categories of this paper are: math-ph
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The categories of this paper are: econ
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The categories of this paper are: q-fin
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The categories of this paper are: atom-ph