Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Decay of the solution to the bipolar Euler-Poisson system with damping
  in $\mathbb{R}^3$   We construct the global solution to the Cauchy's problem of the bipolar
Euler-Poisson equations with damping in $\mathbb{R}^3$ when $H^3$ norm of the
initial data is small. If further, the $\dot{H}^{-s}$ norm ($0\leq s<3/2)$ or
$\dot{B}_{2,\infty}^{-s}$ norm ($0<s\leq3/2$) of the initial data is bounded,
we give the optimal decay rates of the solution. As a byproduct, the decay
results of the $L^p-L^2$ ($1\leq p\leq2$) type hold without the smallness of
the $L^p$ norm of the initial data. In particular, we deduce that
$\|\nabla^k(\rho_1-\rho_2)\|_{L^2} \sim(1+t)^{-5/4-\frac{k}{2}}$ and
$\|\nabla^k(\rho_i-\bar{\rho},u_i,\nabla\phi)\|_{L^2}
\sim(1+t)^{-3/4-\frac{k}{2}}$. We improve the decay results in Li and Yang
\cite{Li3}(\emph{J.Differential Equations} 252(2012), 768-791), where they
showed the decay rates as $\|\nabla^k(\rho_i-\bar{\rho})\|_{L^2}
\sim(1+t)^{-3/4-\frac{k}{2}}$ and $\|\nabla^k(u_i,\nabla\phi)\|_{L^2}
\sim(1+t)^{-1/4-\frac{k}{2}}$, when the $H^3\cap L^1$ norm of the initial data
is small. Our analysis is motivated by the technique developed recently in Guo
and Wang \cite{Guo}(\emph{Comm. Partial Differential Equations} 37(2012),
2165-2208) with some modifications.

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The categories of this paper are: hep-th
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The categories of this paper are: funct-an
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The categories of this paper are: q-alg
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The categories of this paper are: ao-sci
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The categories of this paper are: econ
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The categories of this paper are: mtrl-th
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The categories of this paper are: patt-sol
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The categories of this paper are: comp-gas
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The categories of this paper are: dg-ga
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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: cond-mat
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The categories of this paper are: nlin
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The categories of this paper are: nucl-th
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The categories of this paper are: q-fin
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The categories of this paper are: stat