Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Radial blow-up standing solutions for the semilinear wave equation

Maissâ Boughrara, Hatem Zaag

TL;DR

This work addresses radial blow-up for the semilinear wave equation $\partial_t^2 U=\Delta U+|U|^{p-1}U$ and establishes the existence of blow-up solutions whose similarity profiles converge exponentially to a soliton near a non-characteristic point $(r_0,T(r_0))$. The authors reformulate the problem in self-similar variables, deploy a modulation approach to control unstable directions, and apply energy methods inspired by Merle–Zaag to obtain exponential decay of the error around a generalized soliton, while carefully handling the radial gradient term. A shrinking-set argument traps the modulation parameters and error, yielding convergence in the energy space $\mathcal{H}$ and, via a reverse similarity transform, a blow-up profile on the physical spacetime. The results extend one-dimensional radial insights to higher dimensions outside the origin, provide differentiability of the blow-up curve, and establish a stability framework for the constructed blow-up solution against perturbations, including non-radial ones. The key technical advance is managing the gradient term $e^{-s}(N-1)/(r_0+ye^{-s})\partial_y w$ in the radial setting while preserving the soliton-convergence mechanism.

Abstract

We consider the semilinear wave equation with a power nonlinearity in the radial case. Given $r_0>0$, we construct a blow-up solution such that the solution near $(r_0,T(r_0))$ converges exponentially to a soliton. Moreover, we show that $r_0$ is a non-characteristic point. For that, we translate the question in self-similar variables and use a modulation technique. We will also use energy estimates from the one dimensional case treated by Merle and Zaag in 2007. Of course because of the radial setting, we have an additional gradient term which is delicate to handle. That's precisely the purpose of our paper.

Radial blow-up standing solutions for the semilinear wave equation

TL;DR

This work addresses radial blow-up for the semilinear wave equation and establishes the existence of blow-up solutions whose similarity profiles converge exponentially to a soliton near a non-characteristic point . The authors reformulate the problem in self-similar variables, deploy a modulation approach to control unstable directions, and apply energy methods inspired by Merle–Zaag to obtain exponential decay of the error around a generalized soliton, while carefully handling the radial gradient term. A shrinking-set argument traps the modulation parameters and error, yielding convergence in the energy space and, via a reverse similarity transform, a blow-up profile on the physical spacetime. The results extend one-dimensional radial insights to higher dimensions outside the origin, provide differentiability of the blow-up curve, and establish a stability framework for the constructed blow-up solution against perturbations, including non-radial ones. The key technical advance is managing the gradient term in the radial setting while preserving the soliton-convergence mechanism.

Abstract

We consider the semilinear wave equation with a power nonlinearity in the radial case. Given , we construct a blow-up solution such that the solution near converges exponentially to a soliton. Moreover, we show that is a non-characteristic point. For that, we translate the question in self-similar variables and use a modulation technique. We will also use energy estimates from the one dimensional case treated by Merle and Zaag in 2007. Of course because of the radial setting, we have an additional gradient term which is delicate to handle. That's precisely the purpose of our paper.
Paper Structure (12 sections, 15 theorems, 122 equations)

This paper contains 12 sections, 15 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Explicit solutions
  3. Main results
  4. Setting of the problem
  5. Dynamics in similarity variables
  6. Modulation theory
  7. Energy estimates
  8. Trapping in some shrinking set
  9. Reduction thanks to some shrinking set
  10. Proof of Proposition \ref{['sol conv solition']}
  11. Proof of Theorem \ref{['Thm']}
  12. Toolbox

Key Result

Theorem 1

There exists $\delta\in (0,1)$, such that for all $r_0>0$ and $\hat{d}_0\in (-1,1)$, there exist $s_0> -\log r_0$, $\nu_0, d_0 \in \mathbb{R}$, and $\varepsilon_0> 0$ such that equation eq radial with the initial data given by with $T_0=e^{-s_0}$, has a solution $u(r,t)$ defined for all $(r,t)\in (\mathbb{R}^+)^2$ such that $0\leq t <T(r)$ for some 1-Lipschitz curve $r\mapsto T(r)$. Moreover, $r_

Theorems & Definitions (30)

  • Theorem 1: A solution converging to a soliton in similarity variables
  • Remark 1.1
  • Remark 1.2
  • Corollary 1.3: Convergence in $H^1\times L^2(-1,1)$
  • Definition 1.4: Equivalence of functions in the normalized $L^2_{\{|r-r_0|<T_0-t\}}$ norm
  • Corollary 1.5: Profile in the $u(x,t)$ variables
  • Corollary 1.6: Differentiability and stability with repect to the blow-up point
  • Corollary 1.7: Stability with respect to the initial data in the radial setting
  • Remark 1.8
  • Remark 1.9
  • ...and 20 more