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CPAA

In this paper, we consider the weighted integral system involving
Wolff potentials in $R^{n}$:
\begin{eqnarray}
u(x) = R_1(x)W_{\beta, \gamma}(\frac{u^pv^q(y)}{|y|^\sigma})(x),
\\
v(x) = R_2(x)W_{\beta,\gamma}(\frac{v^pu^q(y)}{|y|^\sigma})(x).
\end{eqnarray}
where $0< R(x) \leq C$, $1 < \gamma \leq 2$, $0\leq \sigma < \beta \gamma$,
$n-\beta\gamma > \sigma(\gamma-1)$,
$\gamma^{*}-1=\frac{n\gamma}{n-\beta\gamma+\sigma}-1\geq 1$. Due to
the weight $\frac{1}{|y|^\sigma}$, we need more complicated
analytical techniques to handle the properties of the solutions.
First, we use the method of regularity lifting to obtain the
integrability for the solutions of this Wolff type integral
equation. Next, we use the modifying and refining method of moving
planes established by Chen and Li to prove the radial symmetry for
the positive solutions of related integral equation. Based on these
results, we obtain the decay rates of the solutions of (0.1) with
$R_1(x)\equiv R_2(x)\equiv 1$ near infinity. We generalize the
results in the related references.

keywords:
Wolff potential
,
regularity lifting
,
integrability
,
method of moving planes
,
decay rates.
,
radial symmetry

DCDS-B

In this paper we prove an optimal-order error estimate for a
family of characteristic mixed method with arbitrary degree of
mixed finite element approximations for the numerical solution of
transient convection diffusion equations. This paper generalizes
the results in [1, 61]. The proof of the main results
is carried out via three lemmas, which are utilized to overcome
the difficulties arising from the combination of MMOC and mixed
finite element methods. Numerical experiments are presented to
justify the theoretical analysis.

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