The authors establish a general monotonicity formula for the following elliptic system △ui + fi(x,u1,··· ,um) = 0 in Ω, where Ω Rn is a regular domain, (fi(x,u1,··· ,um)) = uF(x,u), F(x,u) is a given smooth function of x ∈ Rn and u = (u1,··· ,um) ∈ Rm. 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 tui-△ui-fi(x,u1,··· ,um) = 0 in (t1,t2) × Rn, where t1 < t2 are two constants, (fi(x,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.
