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The authors establish a general monotonicity formula for the following elliptic system     $$\Delta u_i + f_i \left( {x,u_1 , \cdots ,u_m } \right) = 0 in \Omega $$     where Ω ⊂⊂ ℝ n  is a regular domain, (f  i (x, u 1, ..., u  m )) = ∇$$\vec u$$ F(x, $$\vec u$$), F(x, $$\vec u$$) is a given smooth function of x ∈ ℝ n  and $$\vec u$$ = (u 1, ..., u  m ) ∈ 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     $$\partial _t u_i - \Delta u_i - f_i \left( {x,u_1 , \cdots ,u_m } \right) = 0 in \left( {t_1 ,t_2 } \right) \times \mathbb{R}^n $$    , where t 1 t 2 are two constants, (f  i (x, $$\vec u$$)), is given as above. The new monotonicity formulae are focused more attention on the monotonicity of nonlinear terms. The new point of the results is that an index β is introduced to measure the monotonicity of the nonlinear terms in the problems. The index β in the study of monotonicity formulae is useful in understanding the behavior of blow-up sequences of solutions. Another new feature is that the previous monotonicity formulae are extended to nonhomogeneous nonlinearities. As applications, the Ginzburg-Landau model and some different generalizations to the free boundary problems are studied.

New monotonicity formulae for semi-linear elliptic and parabolic systems