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New monotonicity formulae for semi-linear elliptic and parabolic systems

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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 ⊂⊂ 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

$$ \partial _t u_i - \Delta u_i - f_i \left( {x,u_1 , \cdots ,u_m } \right) = 0 in \left( {t_1 ,t_2 } \right) \times \mathbb{R}^n $$

, 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.

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Correspondence to Li Ma.

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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

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  • Elliptic systems
  • Parabolic system
  • Monotonicity formula
  • Ginzburg-Landau model

2000 MR Subject Classification

  • 35Jxx
  • 17B40
  • 17B50