Positive solutions to involving Wolff potentials
Huan-Zhen Chen Zhongxue Lü
Communications on Pure & Applied Analysis 2014, 13(2): 773-788 doi: 10.3934/cpaa.2014.13.773
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
An optimal-order error estimate for a family of characteristic-mixed methods to transient convection-diffusion problems
Huan-Zhen Chen Zhao-Jie Zhou Hong Wang Hong-Ying Man
Discrete & Continuous Dynamical Systems - B 2011, 15(2): 325-341 doi: 10.3934/dcdsb.2011.15.325
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.
keywords: characteristic-mixed methods mixed finite element methods Transient convection diffusion problems optimal order error estimate.

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