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The Cauchy problem for semi-linear Klein-Gordon equations in Friedmann-Lemaître-Robertson-Walker spacetimes

Makoto Nakamura, Takuma Yoshizumi

Abstract

The Cauchy problem for semi-linear Klein-Gordon equations is considered in Friedmann-Lemaître-Robertson-Walker spacetimes. The local and global well-posedness of the Cauchy problem is considered in Sobolev spaces. The non-existence of global solutions is also considered.

The Cauchy problem for semi-linear Klein-Gordon equations in Friedmann-Lemaître-Robertson-Walker spacetimes

Abstract

The Cauchy problem for semi-linear Klein-Gordon equations is considered in Friedmann-Lemaître-Robertson-Walker spacetimes. The local and global well-posedness of the Cauchy problem is considered in Sobolev spaces. The non-existence of global solutions is also considered.

Paper Structure

This paper contains 6 sections, 13 theorems, 190 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['Thm-13-Cor-14']}
  4. Proof of Corollary \ref{['Thm-16-Cor-17']}
  5. Proof of Theorem \ref{['Thm-19']}
  6. Proof of Corollary \ref{['Cor-20']}

Key Result

Theorem 1

Let $n\ge1$. Let $0\le \mu_0<n/2$ if $n=1,2$, and let $0\le \mu_0<n/2-1$ if $n\ge3$. Let $\mu_0\le \mu<\infty$. Let $p$ satisfy Let $\lambda\in \mathbb{C}$, and let $f$ satisfy Def-f. Assume $\mu<p$ unless $f(u)=\lambda |u|^{p-1}u$ for odd $p$, or $f(u)=\lambda |u|^p$ for even $p$. Let $T_0$, $T_1$, $a$ and $M$ satisfy Put $a_0:=a(0)$ and $M_0:=M(0)$. Then the following results hold. (1) (Local

Theorems & Definitions (26)

  • Theorem 1: Local and global solutions
  • Corollary 2: Local and global solutions under \ref{['Def-a']} and \ref{['M-FLRW']}
  • Remark 3: Fujita exponent
  • Theorem 4: Blowing-up under the gauge-invariance
  • Remark 5
  • Corollary 6: Blowing-up under the gauge-invariance under \ref{['Def-a']} and \ref{['M-FLRW']}
  • Lemma 1: Energy estimates
  • proof
  • Lemma 2
  • proof
  • ...and 16 more