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Blow-up solutions of damped Klein-Gordon equation on the Heisenberg group

Michael Ruzhansky, Bolys Sabitbek

Abstract

Inthisnote,weprovetheblow-upofsolutionsofthesemilineardamped Klein-Gordon equation in a finite time for arbitrary positive initial energy on the Heisenberg group. This work complements the paper [21] by the first author and Tokmagambetov, where the global in time well-posedness was proved for the small energy solutions.

Blow-up solutions of damped Klein-Gordon equation on the Heisenberg group

Abstract

Inthisnote,weprovetheblow-upofsolutionsofthesemilineardamped Klein-Gordon equation in a finite time for arbitrary positive initial energy on the Heisenberg group. This work complements the paper [21] by the first author and Tokmagambetov, where the global in time well-posedness was proved for the small energy solutions.
Paper Structure (5 sections, 1 theorem, 50 equations)

This paper contains 5 sections, 1 theorem, 50 equations.

Table of Contents

  1. Introduction
  2. Setting of the problem
  3. Literature overview
  4. Preliminaries on the Heisenberg group
  5. Main Result

Key Result

Theorem 2.1

Let $b>0$, $m>0$ and $\mu= \max\{ b,m, \alpha \}$. Assume that nonlinearity $f(u)$ satisfies where $F(u)$ is as in (Cond-F). Assume that the Cauchy data $u_0 \in H_{\mathcal{L}}^1(\mathbb{H}^n)$ and $u_1 \in L^2(\mathbb{H}^n)$ satisfy and Then the solution of equation (Wave-problem) blows up in finite time $T^*$ such that where the blow-up time $T^* \in (0,T_0)$ with $T_0<+\infty$.

Theorems & Definitions (4)

  • Definition 1.1: Weak solution
  • Theorem 2.1
  • Remark 2.2
  • proof : Proof of Theorem \ref{['thm_main']}