Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Finite time blow-up of semi-linear Klein-Gordon equations with positive initial energy in FLRW spacetimes

Makoto Nakamura, Takuma Yoshizumi

TL;DR

This work studies finite-time blow-up for semi-linear Klein-Gordon equations with positive initial energy in FLRW spacetimes. The authors extend Levine's concavity method from Minkowski space to cosmological backgrounds by introducing an auxiliary concavity function $\theta(t)$ built from the solution energy $E(t)$, the Nehari functional $I(u)$, and the scale factor $a(t)$, deriving explicit blow-up criteria for large data. They establish two main results (Case I and Case II) that yield blow-up in finite time with computable upper bounds on the blow-up time, complemented by corollaries for special scale factors including Minkowski and expanding/de Sitter-like cases. The results improve prior FLRW blow-up criteria by allowing larger initial energy and by refining energy-inequality arguments, thereby advancing understanding of nonlinear wave behavior in cosmological spacetimes and providing practical blow-up predictions. The techniques and conditions developed offer concrete tools for predicting breakdown of solutions under varied cosmological backgrounds.

Abstract

Blowing-up solutions for semi-linear Klein-Gordon equations are considered in Friedmann-Lemaître-Robertson-Walker spacetimes. Some sufficient conditions are shown by applying the concavity method for semi-linear wave equations in the Minkowski spacetime to semi-linear Klein-Gordon equations in FLRW spacetimes.

Finite time blow-up of semi-linear Klein-Gordon equations with positive initial energy in FLRW spacetimes

TL;DR

This work studies finite-time blow-up for semi-linear Klein-Gordon equations with positive initial energy in FLRW spacetimes. The authors extend Levine's concavity method from Minkowski space to cosmological backgrounds by introducing an auxiliary concavity function built from the solution energy , the Nehari functional , and the scale factor , deriving explicit blow-up criteria for large data. They establish two main results (Case I and Case II) that yield blow-up in finite time with computable upper bounds on the blow-up time, complemented by corollaries for special scale factors including Minkowski and expanding/de Sitter-like cases. The results improve prior FLRW blow-up criteria by allowing larger initial energy and by refining energy-inequality arguments, thereby advancing understanding of nonlinear wave behavior in cosmological spacetimes and providing practical blow-up predictions. The techniques and conditions developed offer concrete tools for predicting breakdown of solutions under varied cosmological backgrounds.

Abstract

Blowing-up solutions for semi-linear Klein-Gordon equations are considered in Friedmann-Lemaître-Robertson-Walker spacetimes. Some sufficient conditions are shown by applying the concavity method for semi-linear wave equations in the Minkowski spacetime to semi-linear Klein-Gordon equations in FLRW spacetimes.
Paper Structure (7 sections, 10 theorems, 116 equations, 1 table)

This paper contains 7 sections, 10 theorems, 116 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of theorems and corollaries
  4. Proof of Theorem \ref{['Thm-blow-1']}
  5. Proof of Theorem \ref{['Thm-blow-B']}
  6. Proof of Corollary \ref{['Cor-blowing-up']}
  7. Proof of Corollary \ref{['Cor-blowing-up-B']}

Key Result

Theorem 1

Let $m\in{\mathbb R}$. Let $f\in C({\mathbb C},{\mathbb C})$ be a function with f-condition for some $\varepsilon >0$. Assume that $t_0=0$, and $u_0\in H^1( { \mathbb{R}^n } )$, $u_1\in L^2( { \mathbb{R}^n } )$ satisfy for $E(\cdot)$ given by Def-Energy. Assume $\operatorname{Re} (u_0,u_1)\ge0$. Let $0<T_0\le\infty$. Let $a\in C^2([0,T_0),(0,\infty))$ satisfy $\dot a(t)\ge0$ and for any $t\in[0,

Theorems & Definitions (20)

  • Theorem 1
  • Remark 2
  • Theorem 3
  • Remark 4
  • Corollary 5
  • Corollary 6
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 10 more