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A New Quasi-Singularity Formation Mechanism for Second-order Hyperbolic Equations

Huaian Diao, Xieling Fan, Hongyu Liu

TL;DR

This work introduces a novel quasi-singularity mechanism for second-order hyperbolic equations with anisotropic coefficients: inside a bounded region $D$ the equation can be nonlinear and variable, while outside it reduces to the classical wave equation. By constructing inputs via a transmission-eigenfunction framework and Herglotz wave approximants, the authors force almost-blowup of the spatial gradient near prescribed boundary points without sacrificing well-posedness, for both linear and nonlinear problems and for arbitrary terminal times $T$. A key finding is that the almost-blowup regions have vanishing measure as the amplification target $\mathcal{M}$ grows, balancing higher gradient against shrinking support, and the approach yields $C^{1,1/2}$ regularity near the blow-up sites and $H^1$ or $H^3_{\mathrm{loc}}$ regularity away from $D$. The results rely on a spectral-spatial construction that combines interior transmission eigenfunctions, their Herglotz approximations, and fixed-point arguments to control nonlinear effects over long times, offering a new lens on energy amplification in heterogeneous media with potential implications for inverse problems and wave-control strategies.

Abstract

This paper investigates a novel mechanism for quasi-singularity formation in both linear and nonlinear hyperbolic wave equations in two and three dimensions. We prove that over any finite time interval, there exist inputs such that the Hölder norm of the resulting wave field exceeds any prescribed bound. Conversely, the set of such almost-blowup points has vanishing measure when the aforementioned bound goes to infinity. This phenomenon thus defines a quasi-singular state, intermediate between classical singularity and regularity. Crucially, both the equation coefficients and the inputs can be arbitrarily smooth; the quasi-singularity arises intrinsically from the structure of the hyperbolic wave equation combined with specific input characteristics.

A New Quasi-Singularity Formation Mechanism for Second-order Hyperbolic Equations

TL;DR

This work introduces a novel quasi-singularity mechanism for second-order hyperbolic equations with anisotropic coefficients: inside a bounded region the equation can be nonlinear and variable, while outside it reduces to the classical wave equation. By constructing inputs via a transmission-eigenfunction framework and Herglotz wave approximants, the authors force almost-blowup of the spatial gradient near prescribed boundary points without sacrificing well-posedness, for both linear and nonlinear problems and for arbitrary terminal times . A key finding is that the almost-blowup regions have vanishing measure as the amplification target grows, balancing higher gradient against shrinking support, and the approach yields regularity near the blow-up sites and or regularity away from . The results rely on a spectral-spatial construction that combines interior transmission eigenfunctions, their Herglotz approximations, and fixed-point arguments to control nonlinear effects over long times, offering a new lens on energy amplification in heterogeneous media with potential implications for inverse problems and wave-control strategies.

Abstract

This paper investigates a novel mechanism for quasi-singularity formation in both linear and nonlinear hyperbolic wave equations in two and three dimensions. We prove that over any finite time interval, there exist inputs such that the Hölder norm of the resulting wave field exceeds any prescribed bound. Conversely, the set of such almost-blowup points has vanishing measure when the aforementioned bound goes to infinity. This phenomenon thus defines a quasi-singular state, intermediate between classical singularity and regularity. Crucially, both the equation coefficients and the inputs can be arbitrarily smooth; the quasi-singularity arises intrinsically from the structure of the hyperbolic wave equation combined with specific input characteristics.

Paper Structure

This paper contains 8 sections, 9 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Mathematical setup and summary of major results
  3. Physical motivation and existing studies
  4. Main contributions and technical discussions
  5. Preliminaries
  6. A general framework for specifying inputs
  7. Proof of Theorem \ref{['1.1']}
  8. Proof of Theorem \ref{['thm:main2']}

Key Result

Theorem 1.1

For any arbitrarily large $\mathcal{M} > 0$, $n$ distinct points $\mathbf{x}_1,\mathbf{x}_2,\ldots,\mathbf{x}_n \in \partial D$, and any time interval $[0,T]$ with $0 < T < \infty$, by prescribing appropriate initial and boundary inputs $($as specified in initial and boundary condition$)$, there exi such that problem 1.1 admits a unique solution $u$ satisfying $:$ with the following properties $:$

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Lemma 2.1
  • ...and 20 more