The authors establish a general monotonicity formula for the following elliptic system
where Ω ⊂⊂ ℝn 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 ∈ ℝn and \( \vec u \) = (u 1, ..., u m ) ∈ 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
, 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.
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Project supported by 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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Ma, L., Song, X. & Zhao, L. New monotonicity formulae for semi-linear elliptic and parabolic systems. Chin. Ann. Math. Ser. B 31, 411–432 (2010). https://doi.org/10.1007/s11401-008-0282-8
- Elliptic systems
- Parabolic system
- Monotonicity formula
- Ginzburg-Landau model
2000 MR Subject Classification