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A better bound on blow-up rate for the superconformal semilinear wave equation

Mohamed Ali Hamza, Hatem Zaag

TL;DR

This work advances the understanding of finite-time blow-up for the higher-dimensional semilinear wave equation with superconformal nonlinearity by proving a new upper bound on the blow-up rate that gains a $|\log(T-t)|^q$ factor over previous results. The authors leverage a self-similar variable framework and develop a suite of Lyapunov functionals, built from Pohozaev-type multipliers and weighted energy identities in the similarity variables, to obtain uniform space-time bounds and a hierarchical decay structure. A key contribution is the construction of a family of Lyapunov functionals $\mathcal{F}_k$ via an inductive scheme, which couples with a weighted bound on the $L^{p+1}$ norm of the similarity profile $w$ to drive increasingly strong decay estimates. These mechanisms culminate in a log-enhanced blow-up-rate bound at non-characteristic points, with the proof organized around a sequence of energy identities and monotonicity formulas, and supported by precise elementary identities for the Pohozaev multipliers. The results sharpen existing blow-up rate bounds in the subcritical, Sobolev-subcritical regime and offer a robust framework for further refinements in the superconformal range.

Abstract

We consider the semilinear wave equation in higher dimensions with superconformal power nonlinearity. The purpose of this paper is to give a new upper bound on the blow-up rate in some space-time integral, showing a $|\log(T-t)|^q$ improvement in comparison with previous results obtained in \cite{HZdcds13,KSVsurc12}.

A better bound on blow-up rate for the superconformal semilinear wave equation

TL;DR

This work advances the understanding of finite-time blow-up for the higher-dimensional semilinear wave equation with superconformal nonlinearity by proving a new upper bound on the blow-up rate that gains a factor over previous results. The authors leverage a self-similar variable framework and develop a suite of Lyapunov functionals, built from Pohozaev-type multipliers and weighted energy identities in the similarity variables, to obtain uniform space-time bounds and a hierarchical decay structure. A key contribution is the construction of a family of Lyapunov functionals via an inductive scheme, which couples with a weighted bound on the norm of the similarity profile to drive increasingly strong decay estimates. These mechanisms culminate in a log-enhanced blow-up-rate bound at non-characteristic points, with the proof organized around a sequence of energy identities and monotonicity formulas, and supported by precise elementary identities for the Pohozaev multipliers. The results sharpen existing blow-up rate bounds in the subcritical, Sobolev-subcritical regime and offer a robust framework for further refinements in the superconformal range.

Abstract

We consider the semilinear wave equation in higher dimensions with superconformal power nonlinearity. The purpose of this paper is to give a new upper bound on the blow-up rate in some space-time integral, showing a improvement in comparison with previous results obtained in \cite{HZdcds13,KSVsurc12}.
Paper Structure (15 sections, 228 equations)

This paper contains 15 sections, 228 equations.

Table of Contents

  1. Introduction
  2. Motivation of the problem
  3. Strategy of the proof
  4. A bound to the time average of the $L^{p+1}$ norm of $w$ with singular weight
  5. Classical energy estimates
  6. New energy estimates
  7. Proof of Proposition \ref{['P01m']}':
  8. Improvments about the $\dot{H}^{1}({\mathbb {R}}^N\times [-\log(s_0),\infty))$ norm of $w$
  9. Proof of Proposition \ref{['improv']}
  10. Slightly more general improvments about the norm of $w$ in the space $\dot{H}^{1}({\mathbb {R}}^N\times [-\log(s_0),\infty))$ under assumptions
  11. Proof of Theorem \ref{['t0']}
  12. Existence of a Lyapunov functional for equation (\ref{['C']})
  13. Proof of Theorem \ref{['t0']} in the particular case $q=\frac{1}{18}$
  14. Proof of Theorem \ref{['t0']}
  15. Some elementary identities

Theorems & Definitions (5)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5