Mathematics – Analysis of PDEs
Scientific paper
2005-10-10
Mathematics
Analysis of PDEs
38 pages
Scientific paper
In this paper, we establish a general monotonicity formula of the following elliptic system $$ \Delta u_i+f_i(u_1,...,u_m)=0 \quad {\rm in} \Omega, \label{0.1} $$ where $\Omega\subset\subset \mathbb{R}^n$ is a bounded domain, $(f_i(u_1,...,u_m))=\nabla F(\vec{u})$, and $F(\vec{u})$ is a given smooth function of $\vec{u}=(u_1,...,u_m)$, $m,n$ are two positive integers. We also set up a new monotonicity formula for the following parabolic system $$ \partial_t u_i-\Delta u_i-f_i(u_1,...,u_m)=0, in (t_1, t_2)\times \mathbb{R}^n, $$ where $t_1
Ma Liangping
Song Xianfa
Zhao Lin
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