New monotonicity formulae for semi-linear elliptic and parabolic systems

Abstract

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.

Keywords

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

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