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Article overview
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Sharp asymptotics for the minimal mass blow up solution of critical gKdV equation | | Vianney Combet
; Yvan Martel
; | | Date: |
10 Feb 2016 | | Abstract: | Let $S$ be a minimal mass blow up solution of the critical generalized KdV
equation as constructed by Martel, Merle and Rapha"el in arXiv:1204.4624. We
prove both time and space sharp asymptotics for $S$ close to the blow up time.
Let $Q$ be the unique ground state of (gKdV), satisfying $Q"+Q^5=Q$.
First, we show that there exist universal smooth profiles
$Q_kinmathcal{S}(mathbb{R})$ (with $Q_0=Q$) and a constant
$c_0inmathbb{R}$ such that, fixing the blow up time at $t=0$ and appropriate
scaling and translation parameters, $S$ satisfies, for any $mgeqslant 0$, [
partial_x^m S(t) - sum_{k=0}^{[m/2]} frac 1{t^{frac 12+m-2k}}
Q_k^{(m-k)}left(frac{cdot+ frac1t}{t}+c_0
ight) o 0quad mbox{in} L^2
mbox{as} tdownarrow 0. ] Second, we prove that, for $0<tll 1$, $xleqslant
-frac 1t -1$, [ S(t,x) sim - frac 12 |Q|_{L^1} |x|^{-3/2}, ] and related
bounds for the derivatives of $S(t)$ of any order. We also prove
$int_{mathbb{R}} S(t,x),dx=0$. | | Source: | arXiv, 1602.3519 | | Services: | Forum | Review | PDF | Favorites | |
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