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Standing wave solutions and instability for the Logarithmic Klein-Gordon equation

Lijia Han, Yue Qiu, Xiaohong Wang

Abstract

In this paper, we study the standing wave solutions of Klein--Gordon equation with logarithmic nonlinearity. The existence of the standing wave solution related to the ground state $φ_0(x)$ is obtained. Further, we prove the instability of solutions around $φ_0(x)$.

Standing wave solutions and instability for the Logarithmic Klein-Gordon equation

Abstract

In this paper, we study the standing wave solutions of Klein--Gordon equation with logarithmic nonlinearity. The existence of the standing wave solution related to the ground state is obtained. Further, we prove the instability of solutions around .
Paper Structure (6 sections, 8 theorems, 98 equations)

This paper contains 6 sections, 8 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and statement of main results
  3. Notations
  4. The main results
  5. Existence of Standing Wave Solutions
  6. The Instability of Standing Wave Solutions

Key Result

Theorem 2.1

Let $2<p<4$, $N=3$ and $\omega \in[0,1]$, (i) variational problem are equivalent to minimization problems (ii) there exists $\phi_0 \in M_\omega$ such that $d(\omega)=\underset{\phi \in M_\omega}{\inf} J_\omega(\phi)=J_\omega\left(\phi_0\right)$, (iii) $\phi_0$ is a solution of $-\Delta \phi+\left(1-\omega^2\right) \phi=|\phi|^{p-1} \phi \ln |\phi|^2$.

Theorems & Definitions (12)

  • Theorem 2.1
  • Theorem 2.2
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Corollary 3.1
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 2 more