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Nonexistence of global weak solutions of Klein-Gordon equations with gauge variant semilinear terms in Friedmann-Lemaître-Robertson-Walker spacetimes

Makoto Nakamura, Takuma Yoshizumi

TL;DR

The paper establishes the nonexistence of global weak solutions for gauge-variant semilinear Klein-Gordon equations on FLRW spacetimes, extending blow-up results to cosmological backgrounds with expanding, contracting, or singular scale functions. It develops a test-function framework adapted to curved backgrounds, reducing the problem to an ODE-type inequality for $w(t)=\operatorname{Re}\int_{\mathbb{R}^n} u(t,x)\,dx$ and leveraging finite speed of propagation to obtain a contradiction under precise growth and mass conditions. The main contributions include a rigorous nonexistence result for a broad class of masses (including purely imaginary) and nonlinearities with $1<p<\infty$, and a detailed corollary that covers concrete FLRW models (Big-Rip/Big-Crunch) with explicit $p$-range conditions. The findings enhance understanding of dissipative effects induced by cosmological expansion/contraction on semilinear Klein-Gordon dynamics and provide groundwork for further stability/blow-up analyses in curved spacetimes.

Abstract

Nonexistence of global weak solutions of Klein-Gordon equations with gauge variant semilinear terms are considered in Friedmann-Lemaître-Robertson-Walker spacetimes. Effects of spatial expansion or contraction on the solutions are studied through the scale-function and the curved mass.

Nonexistence of global weak solutions of Klein-Gordon equations with gauge variant semilinear terms in Friedmann-Lemaître-Robertson-Walker spacetimes

TL;DR

The paper establishes the nonexistence of global weak solutions for gauge-variant semilinear Klein-Gordon equations on FLRW spacetimes, extending blow-up results to cosmological backgrounds with expanding, contracting, or singular scale functions. It develops a test-function framework adapted to curved backgrounds, reducing the problem to an ODE-type inequality for and leveraging finite speed of propagation to obtain a contradiction under precise growth and mass conditions. The main contributions include a rigorous nonexistence result for a broad class of masses (including purely imaginary) and nonlinearities with , and a detailed corollary that covers concrete FLRW models (Big-Rip/Big-Crunch) with explicit -range conditions. The findings enhance understanding of dissipative effects induced by cosmological expansion/contraction on semilinear Klein-Gordon dynamics and provide groundwork for further stability/blow-up analyses in curved spacetimes.

Abstract

Nonexistence of global weak solutions of Klein-Gordon equations with gauge variant semilinear terms are considered in Friedmann-Lemaître-Robertson-Walker spacetimes. Effects of spatial expansion or contraction on the solutions are studied through the scale-function and the curved mass.

Paper Structure

This paper contains 3 sections, 4 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['Thm-2']}
  3. Proof of Corollary \ref{['Cor-3']}

Key Result

Theorem 2

Let $n\ge1$, $\lambda>0$, $1<p<\infty$ and $r_0>0$. Let Put Let $a\in C([0,\infty),(0,\infty))$, $M\in C([0,\infty),i{\mathbb R})$, and let $N$ be a number with Let $0<\theta<1$. Put where $\omega_n$ denotes the volume of unit ball in ${ \mathbb{R}^n }$, and assume Then the nontrivial weak solution $u$ of Cauchy does not exist.

Theorems & Definitions (8)

  • Definition 1: Weak global solution
  • Theorem 2: Nonexistence of global weak solutions
  • Corollary 3: Nonexistence of global weak solutions under \ref{['Def-a']}
  • Remark 4
  • Remark 5
  • Lemma 1
  • Lemma 1
  • Remark 2