Long-Time Dispersive Estimates for Perturbations of a Kink Solution of One-Dimensional Cubic Wave Equations

Abstract

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Quadratic lifespan and growth of Sobolev norms for derivative Schrödinger equations on generic tori

“…Hence we are only able to obtain the bound (1.7) over a long (but finite) time interval, by constructing a modified energy for the H ρ -norm with normal forms techniques. The result we obtained is in the same spirit of [20] by Delort-Masmoudi. We require the Hamiltonian assumption on the nonlinearity in order to guarantee the wellposedness of the Cauchy problem associated to (1.1) at least for short time. Actually this hypothesis could be weakened.…”

Dynamics of kink clusters for scalar fields in dimension 1+1

“…Let us stress again that the main object of our study are solutions approaching multisoliton configurations in the strong energy norm, in other words we address the question of interaction of solitons in the absence of radiation. Allowing for a radiation term seems to be currently out of reach, the question of the asymptotic stability of the kink being still unresolved, see for example [9,24,13,16,3,28] for recent results on this and related problems.…”

Quadratic Klein-Gordon equations with a potential in one dimension

“…Full asymptotic stability for kinks of relativistic GL equations (1.24) was proven by Komech-Kopylova [41,42] when 𝑝 ≥ 13. In a very recent paper, Delort and Masmoudi [13] proved long time stability for the kink of the 𝜙 4 model, reaching times of order 𝜖 −4 for data of size 𝜖; their analysis is based on a semi-classical approach using conjugation by the wave operators. Concerning this last problem, as a consequence of our general results on equation (KG), we can obtain a global stability result (in the odd class) provided the nonlinearity is projected onto the continuous spectrum.…”

“…Since 𝑉 0 and its zero energy resonance are even, our results apply to show global bounds and decay for equation (1.19) with odd data. Thus, we are able to settle at least part of the kink stability problem; the remaining difficulty, in the odd case, is to prove that the coupling of the internal mode to the continuous spectrum causes the energy of the internal mode to be dispersed through the phenomenon of 'radiation damping' [69,13]. This is a serious obstacle since the presence of the internal mode leads to the formation of a singularity in distorted Fourier space, at the frequency given by the Fermi golden rule.…”

“…13 gives (using |𝜃| |𝜎|)C 𝑐𝑟𝑟 ( 𝑓 , 𝑓 , 𝑓 ) 𝐿 2 𝑡 −1 • 𝜒 𝑐 (𝜂)𝜕 𝜂 𝑓 𝐿 2+ 𝜕 𝑥 0+ 𝑒 −𝑖𝑡 𝜕 𝑥 F −1 𝜕 𝜎 𝑓 𝐿 ∞− • 𝜕 𝑥 0+ 𝑒 −𝑖𝑡 𝜕 𝑥 W * 𝑓 𝐿 ∞− 𝑡 −1 𝜒 𝑐 (𝜂)𝜕 𝜂 𝑓 𝐿 2 𝜒 𝑟 (𝜎) 𝜎 𝜕 𝜎 𝑓 𝐿 2 𝜕 𝑥 0+ 𝑒 −𝑖𝑡 𝜕 𝑥 W * 𝑓 𝐿 ∞− 𝑡 −1 • 𝜀 1 𝑡 𝛼+𝛽𝛾 • 𝜀 1 𝑡 𝛼 • 𝜀 1 𝑡 −1/2+linear decay and the 𝐻 4 -norm, the a priori bounds (see in particular equation (7.19)) and 𝛼 + 𝛽𝛾 < 1/4.…”