On the focusing critical semi-linear wave equation
نویسندگان
چکیده
منابع مشابه
On the Focusing Critical Semi-linear Wave Equation
The wave equation ∂ttψ −∆ψ − ψ 5 = 0 in R is known to exhibit finite time blowup for large data. It also admits the special static solutions φ(x, a) = (3a) 1 4 (1 + a|x|) 1 2 for all a > 0 which are linearly unstable. We view these functions as a curve in the energy space Ḣ × L. We show that in a small neighborhood of itself, which lies on a stable hyper-surface of radial data, this curve acts ...
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Given ν > 1 2 and δ > 0 arbitrary, we prove the existence of energy solutions of (0.1) ∂ttu−∆u− u = 0 in R3+1 that blow up exactly at r = t = 0 as t → 0−. These solutions are radial and of the form u = λ(t) 1 2W (λ(t)r)+η(r, t) inside the cone r ≤ t, where λ(t) = t−1−ν , W (r) = (1 + r2/3)− 1 2 is the stationary solution of (0.1), and η is a radiation term with Z [r≤t] ` |∇η(x, t)| + |ηt(x, t)|...
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We survey recent results related to soliton resolution.
متن کاملGlobal Well-posedness, Scattering and Blow-up for the Energy Critical Focusing Non-linear Wave Equation
In this paper we consider the energy critical non-linear wave equation ∂ t u−∆u = ± |u| 4 N−2 u (x, t) ∈ R × R u ∣∣ t=0 = u0 ∈ Ḣ1(R ) ∂tu ∣∣ t=0 = u1 ∈ L(R ) Here the − sign corresponds to the defocusing problem, while the + sign corresponds to the focusing problem. The theory of the local Cauchy problem (CP) for this equation was developed in many papers, see for instance [26], [9], [2...
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ژورنال
عنوان ژورنال: American Journal of Mathematics
سال: 2007
ISSN: 1080-6377
DOI: 10.1353/ajm.2007.0021