Profiles and Quantization of the Blow Up Mass for Critical Nonlinear Schrödinger Equation
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"Profiles and Quantization of the Bl..." refers background in this paper
...(4) Equation (4) is a standard nonlinear elliptic equation, and from [1] and [6], there is a unique positive solution up to translation Qω(x)....
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...Let ( x−x(t) R(t0) ) ∈ Supp(ψ) ⊂ [1, 2], then:∣∣∣∣x− x(t)R(t0) ∣∣∣∣ ≥ 1 implies ∣∣∣∣x− x(t)λ(t) ∣∣∣∣ = ∣∣∣∣x− x(t)R(t0) ∣∣∣∣ R(t0)λ(t) ≥ 100b(t) ....
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...Let ( x−x(t) R(t0) ) ∈ Supp(ψ) ⊂ [1, 2], then: ∣∣∣∣x− x(t) R(t0) ∣∣∣∣ ≥ 1 implies ∣∣∣∣x− x(t) λ(t) ∣∣∣∣ = ∣∣∣∣x− x(t) R(t0) ∣∣∣∣ R(t0) λ(t) ≥ 100 b(t) ....
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...(78) Remark 9 From now on, parameter η is fixed small enough so that above estimates hold, and so is a = √ η. Proof of Lemma 4 We first claim from support property of ψ: ∀t ∈ [t0, T ), QS(t, x) = 0 ie u(t, x) = ũ(t, x) for x− x(t) R(t0) ∈ Supp(ψ)....
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"Profiles and Quantization of the Bl..." refers background or result in this paper
...The behavior in R of the right-hand side is a consequence of (74), or more specifically as exhibited in [14] by the fact that in the region |y |≤ A(t), a good approximation of the solution e(t) is the universal radiation ζb(t) of Lemma 1. From (75), we now may estimate the size of the region in space for which this approximation is good, which will turn out to be much smaller than the one suggested at a formal level in [ 19 ]....
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...• On the other hand, numerical simulations, [7], and formal arguments, [ 19 ], suggest...
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