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A Blowup Solution of Multispeed Klein-Gordon System in Space Dimension Two with Small Initial Data

Xilu Zhu

Abstract

We find an example to illustrate that the first nondegeneracy condition of (1.2) is actually needed in proving the global exsitence of 2D multispeed Klein-Gordon system with small initial data (See [3]). We construct a collection of special Klein-Gordon dispersive relations and by iterating the corresponding profiles we can find a blowup in finite time.

A Blowup Solution of Multispeed Klein-Gordon System in Space Dimension Two with Small Initial Data

Abstract

We find an example to illustrate that the first nondegeneracy condition of (1.2) is actually needed in proving the global exsitence of 2D multispeed Klein-Gordon system with small initial data (See [3]). We construct a collection of special Klein-Gordon dispersive relations and by iterating the corresponding profiles we can find a blowup in finite time.
Paper Structure (3 sections, 10 theorems, 162 equations)

This paper contains 3 sections, 10 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Notations and Lemmas
  3. Proof of the Main Theorem

Key Result

Theorem 1.1

Consider the system (1.5) in $\left[0,+\infty\right)\times\mathbb{R}_x^2$. Then for any arbitrarily small $\varepsilon>0$, there exists an initial condition $\textbf{u}_0$ such that (i) If $l\le 0$, then $\widehat{P_l\,\textbf{u}_0} \neq 0$; (ii) $\left\|\textbf{u}_0\right\|_{L^2}\lesssim \varepsilo

Theorems & Definitions (23)

  • Theorem 1.1
  • proof
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • Corollary 2.3
  • proof
  • Proposition 3.1
  • ...and 13 more