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Existence and blow up of small-amplitude nonlinear waves with a sign-changing potential

Paschalis Karageorgis

TL;DR

The paper analyzes the nonlinear wave equation with a sign-changing potential in any space dimension, focusing on when small initial data yield long-lived small-amplitude solutions versus finite-time blow-up depending on the potential and the linearized operator; specifically, if the potential is small and rapidly decaying, the nonlinear term governs existence, whereas growth in $-\Delta+V$ can trigger blow-up. The authors employ radial reduction and a fixed-point framework using the Duhamel operator, along with a refined analysis of the radial-derivative bounds of the Riemann operator, to construct a contraction in a weighted Banach space and obtain a unique solution (local or extended in time) via a Picard iteration, with Lifespan results indicating that in the supercritical case the existence time can be arbitrarily large and, in the subcritical case, $T \ge C \varepsilon^{-(p-1)/(2-k(p-1))}$. A subcritical decay result yields finite-time blow-up by deriving a differential inequality $f''(t)-\lambda f(t) \ge C|f(t)|^p$ with $f(t)=\int \chi u\,dx$, exploiting positivity of the Riemann operator and spectral properties of $-\Delta+V$; the work also discusses when $-\Delta+V$ has negative eigenvalues and the exponential decay of corresponding eigenfunctions. Overall, the results clarify how the sign and decay of the potential govern the transition between prolonged small-amplitude solutions and finite-time blow-up, providing existence, uniqueness, and blow-up analyses along with explicit lifespan estimates.

Abstract

We study the nonlinear wave equation with a sign-changing potential in any space dimension. If the potential is small and rapidly decaying, then the existence of small-amplitude solutions is driven by the nonlinear term. If the potential induces growth in the linearized problem, however, solutions that start out small may blow-up in finite time.

Existence and blow up of small-amplitude nonlinear waves with a sign-changing potential

TL;DR

The paper analyzes the nonlinear wave equation with a sign-changing potential in any space dimension, focusing on when small initial data yield long-lived small-amplitude solutions versus finite-time blow-up depending on the potential and the linearized operator; specifically, if the potential is small and rapidly decaying, the nonlinear term governs existence, whereas growth in can trigger blow-up. The authors employ radial reduction and a fixed-point framework using the Duhamel operator, along with a refined analysis of the radial-derivative bounds of the Riemann operator, to construct a contraction in a weighted Banach space and obtain a unique solution (local or extended in time) via a Picard iteration, with Lifespan results indicating that in the supercritical case the existence time can be arbitrarily large and, in the subcritical case, . A subcritical decay result yields finite-time blow-up by deriving a differential inequality with , exploiting positivity of the Riemann operator and spectral properties of ; the work also discusses when has negative eigenvalues and the exponential decay of corresponding eigenfunctions. Overall, the results clarify how the sign and decay of the potential govern the transition between prolonged small-amplitude solutions and finite-time blow-up, providing existence, uniqueness, and blow-up analyses along with explicit lifespan estimates.

Abstract

We study the nonlinear wave equation with a sign-changing potential in any space dimension. If the potential is small and rapidly decaying, then the existence of small-amplitude solutions is driven by the nonlinear term. If the potential induces growth in the linearized problem, however, solutions that start out small may blow-up in finite time.

Paper Structure

This paper contains 7 sections, 31 theorems, 266 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A Priori Estimates
  4. Existence of solutions
  5. Blow-up due to the initial data
  6. Blow-up due to the Potential
  7. The Riemann Operator in the Radial Case

Key Result

Theorem 1.1

Let n\geq 4 and define m by m. Suppose \varphi\in \mathcal{C}^2({\mathbb R}_+) and \psi\in \mathcal{C}^1({\mathbb R}_+) are subject to da3 for some \varepsilon>0 and k\geq 0. Now, consider the nonlinear wave equation with potential nl. Suppose the nonlinear term F(u) satisfies F for some where p_n is the positive root of the quadratic pn. Also, assume the potential term V(r) is subject to V for s

Figures (1)

  • Figure 1: Our comparison lemma applies only in the shaded region $\Theta_T'$.

Theorems & Definitions (39)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Lemma 2.1
  • Lemma 2.2
  • Proposition 2.3
  • ...and 29 more