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| New Monotonicity Formulae for Semi-linear Elliptic and Parabolic Systems |
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Citation： |
Li MA,Xianfa SONG,Lin ZHAO.New Monotonicity Formulae for Semi-linear Elliptic and Parabolic Systems[J].Chinese Annals of Mathematics B,2010,31(3):411~432 |
| Page view： 1721
Net amount： 1210 |
Authors： |
Li MA; Xianfa SONG; Lin ZHAO; |
Foundation： |
the National Natural Science Foundation of China (No. 10631020) and the Specialized
Research Fund for the Doctoral Program of Higher Education (No. 20060003002). |
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| Abstract： |
The authors establish a general monotonicity formula for the
following elliptic system
$$\Delta u_i+f_i(x,u_1,\cdots,u_m)=0\quad\mbox{in }\Omega,$$
where $\Omega\subset\subset \mathbb{R}^n$ is a regular domain,
$(f_i(x,u_1,\cdots,u_m))=\nabla_{\vec{u}} F(x,\vec{u})$,
$F(x,\vec{u})$ is a given smooth function of $x\in\mathbb{R}^n$ and
$\vec{u}=(u_1,\cdots,u_m)\in\mathbb{R}^m$. The system comes from
understanding the stationary case of Ginzburg-Landau model. A new
monotonicity formula is also set up for the following parabolic
system
$$\partial_t u_i-\Delta u_i-f_i(x,u_1,\cdots,u_m)=0\quad\mbox{in }(t_1, t_2)\times\mathbb{R}^n,$$
where $t_1 |
Keywords： |
Elliptic systems, Parabolic system, Monotonicity formula,
Ginzburg-Landau model |
Classification： |
35Jxx, 17B40, 17B50 |
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