Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Spreading Speeds and Traveling waves of a parabolic-elliptic chemotaxis
  system with logistic source on R^N   In this paper, we study the spreading speeds and traveling wave solutions of
the PDE $$ \begin{cases} u_{t}= \Delta u-\chi \nabla \cdot (u \nabla v) +
u(1-u),\ \ x\in\mathbb{R}^N 0=\Delta v-v+u, \ \ x\in\mathbb{R}^N, \end{cases}
$$ where $u(x,t)$ and $v(x,t)$ represent the population and the chemoattractant
densities, respectively, and $\chi$ is the chemotaxis sensitivity. It has been
shown in an earlier work by the authors of the current paper that, when
$0<\chi<1$, for every nonnegative uniformly continuous and bounded function
$u_0(x)$, the system has a unique globally bounded classical solution
$(u(x,t;u_0),v(x,t;u_0))$ with initial condition $u(x,0;u_0)=u_0(x)$.
Furthermore, if $0<\chi<\frac{1}{2}$, then the constant steady-state solution
$(1,1)$ is asymptotically stable with respect to strictly positive
perturbations. In the current paper, we show that if $0<\chi<1$, then there are
nonnegative constants $c_{-}^*(\chi)\leq c_+^*(\chi)$ such that for every
nonnegative initial function $u_0(\cdot)$ with nonempty compact support $$
\lim_{t\to\infty} \sup_{|x|\le ct} \big[|u(x,t;u_0)-1|+|v(x,t;u_0)-1|\big]=0\ \
\forall\ 0<c<c_{-}^*(\chi) $$ and $$ \lim_{t\to\infty}\sup_{|x|\geq ct} \big[
u(x,t;u_0)+v(x,t;u_0)\big]=0\ \ \forall\ c>c_{+}^*(\chi).$$ We also show that
if $0<\chi<\frac{1}{2}$, there is a positive constant $c^*(\chi)$ such that for
every $c\ge c^*(\chi)$, the system has a traveling wave solution
$(u(x,t),v(x,t))$ with speed $c$ and connecting $(1,1)$ and $(0,0)$, that is,
$(u(x,t),v(x,t))=(U(x-ct),V(x-ct))$ for some functions $U(\cdot)$ and
$V(\cdot)$ satisfying $(U(-\infty),V(-\infty))=(1,1)$ and
$(U(\infty),V(\infty))=(0,0)$. Moreover, $$ \lim_{\chi\to
0}c^*(\chi)=\lim_{\chi\to 0}c_+^*(\chi)=\lim_{\chi\to 0}c_-^*(\chi)=2. $$ We
first give a detailed study in the case $N=1$ and next we extend these results
to the case $N\ge 2$.

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The categories of this paper are: astro-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: gr-qc
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The categories of this paper are: econ
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The categories of this paper are: funct-an
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The categories of this paper are: solv-int
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The categories of this paper are: atom-ph
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The categories of this paper are: q-fin
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-ex
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The categories of this paper are: hep-lat
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The categories of this paper are: nucl-th
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The categories of this paper are: chem-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-alg
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The categories of this paper are: adap-org