Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Full range of infinite point blow-up exponents for the critical generalized KdV equation

Nailya Manatova

TL;DR

This work proves that the mass-critical gKdV equation admits finite-time blow-up with an infinite-point concentration for every blow-up exponent ν in (1/2,1). By inserting a carefully designed right-hand tail f with decay θ>1 into a refined soliton framework, and by a novel mixed virial-energy functional tailored to the scaling of the problem, the authors construct solutions closely following a modulated Q_b profile. The modulation laws yield λ(s)~s^{−β}, σ(s)~s^{1−β} with β=2(θ−1)/(2θ−1), which in turn produce the blow-up rate ∥∂_xU(t)∥_{L^2}~(T−t)^{−ν} for ν=β/(3β−1)∈(1/2,1). The result extends previous exotic blow-up constructions by covering a full continuum of rates above the critical value and highlights the tail–soliton interaction as the mechanism driving infinite-point blow-up; the method hinges on a refined profile Q_b, a scaling-aware bootstrap, and a monotone Lyapunov functional to control the error terms. Overall, the paper completes the picture of non-generic finite-time blow-up for the L^2-critical gKdV by demonstrating a continuum of infinite-point blow-up exponents and clarifying the role of the tail in shaping the dynamics.

Abstract

For the quintic, mass critical generalized Korteweg-de Vries equation, for any $ν\in (\frac{1}{2}, 1)$, we prove the existence of solutions in the energy space that blow up in finite time $T>0$ with the blow-up rate $\|\partial_x u(t)\|_{L^2} \sim (T-t)^{-ν}$ (infinite point blow-up). These solutions are constructed arbitrarily close to the family of solitons and correspond to the concentration of a soliton traveling at $+\infty$ in space as $t\uparrow T$. This complements the previous results obtained in the work of Martel, Merle, Raphaël in 2015 on infinite point exotic blow-up, which were valid under the technical restriction $ν>\frac {11}{13}$. The value $ν=\frac 12$ corresponds to a critical case to be treated elsewhere. At the technical level, we implement a modification of the virial-energy functional, to allow all $ν> \frac 12$ and simplify the proof of energy estimates.

Full range of infinite point blow-up exponents for the critical generalized KdV equation

TL;DR

This work proves that the mass-critical gKdV equation admits finite-time blow-up with an infinite-point concentration for every blow-up exponent ν in (1/2,1). By inserting a carefully designed right-hand tail f with decay θ>1 into a refined soliton framework, and by a novel mixed virial-energy functional tailored to the scaling of the problem, the authors construct solutions closely following a modulated Q_b profile. The modulation laws yield λ(s)~s^{−β}, σ(s)~s^{1−β} with β=2(θ−1)/(2θ−1), which in turn produce the blow-up rate ∥∂_xU(t)∥_{L^2}~(T−t)^{−ν} for ν=β/(3β−1)∈(1/2,1). The result extends previous exotic blow-up constructions by covering a full continuum of rates above the critical value and highlights the tail–soliton interaction as the mechanism driving infinite-point blow-up; the method hinges on a refined profile Q_b, a scaling-aware bootstrap, and a monotone Lyapunov functional to control the error terms. Overall, the paper completes the picture of non-generic finite-time blow-up for the L^2-critical gKdV by demonstrating a continuum of infinite-point blow-up exponents and clarifying the role of the tail in shaping the dynamics.

Abstract

For the quintic, mass critical generalized Korteweg-de Vries equation, for any , we prove the existence of solutions in the energy space that blow up in finite time with the blow-up rate (infinite point blow-up). These solutions are constructed arbitrarily close to the family of solitons and correspond to the concentration of a soliton traveling at in space as . This complements the previous results obtained in the work of Martel, Merle, Raphaël in 2015 on infinite point exotic blow-up, which were valid under the technical restriction . The value corresponds to a critical case to be treated elsewhere. At the technical level, we implement a modification of the virial-energy functional, to allow all and simplify the proof of energy estimates.

Paper Structure

This paper contains 15 sections, 18 theorems, 312 equations.

Table of Contents

  1. Introduction
  2. Presentation of the problem
  3. Main result
  4. Some related results for other nonlinear dispersive or wave models
  5. Notation
  6. Strategy of the proof
  7. Definition of the tail on the right-hand side
  8. Decomposition around the soliton
  9. Structure of the linearized operator
  10. Definitions and estimates of localized profiles
  11. Definition of the approximate solution
  12. Modulation and parameters estimates
  13. Bootstrap estimates
  14. Energy estimates
  15. Construction of blowing up solutions

Key Result

Theorem I.1

Let $\nu \in (\frac{1}{2}, 1)$. For any $\delta >0$, there exist $T>0$ and $U_0 \in H^1(\mathbb{R})$ with $\|U_0 - Q\|_{H^1} \leq \delta$ such that the $H^1$ solution $U$ of gKdV_principal_eq with initial data $U_0$ exists on $[0,T)$ and blows up at the finite time $T$ with

Theorems & Definitions (42)

  • Theorem I.1
  • Lemma II.1: MMPIII
  • proof
  • Remark II.1
  • Lemma III.1: Properties of the linearised operator $\mathcal{L}$
  • Remark III.1
  • Definition III.1: Localised profile
  • Lemma III.2: Approximate self-similar profiles $Q_b$
  • proof
  • Lemma III.3
  • ...and 32 more