Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Blow-up, decay, and convergence to equilibrium for focusing damped cubic Klein-Gordon and Duffing equations

Thomas Perrin

Abstract

We study long-time dynamics of the damped focusing cubic Klein-Gordon equation on a compact three-dimensional Riemannian manifold, together with its space-independent reduction, the damped focusing Duffing equation. Under the geometric control condition on the damping and an assumption on the set of stationary solutions, we establish a sharp trichotomy for initial data with energy slightly above that of the ground state: every solution either blows up in finite time, decays exponentially to zero, or converges to a ground state. We provide a complete classification of the Duffing dynamics above the energy of the constant solution, use it to construct Klein-Gordon solutions realising each of the three behaviours, and derive a simple spectral criterion ensuring that the ground states are nonconstant--and hence that the solutions realising the three regimes lie above the ground-state energy.

Blow-up, decay, and convergence to equilibrium for focusing damped cubic Klein-Gordon and Duffing equations

Abstract

We study long-time dynamics of the damped focusing cubic Klein-Gordon equation on a compact three-dimensional Riemannian manifold, together with its space-independent reduction, the damped focusing Duffing equation. Under the geometric control condition on the damping and an assumption on the set of stationary solutions, we establish a sharp trichotomy for initial data with energy slightly above that of the ground state: every solution either blows up in finite time, decays exponentially to zero, or converges to a ground state. We provide a complete classification of the Duffing dynamics above the energy of the constant solution, use it to construct Klein-Gordon solutions realising each of the three behaviours, and derive a simple spectral criterion ensuring that the ground states are nonconstant--and hence that the solutions realising the three regimes lie above the ground-state energy.

Paper Structure

This paper contains 35 sections, 16 theorems, 150 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Definitions and notations for \ref{['duffing']}
  3. Main results for \ref{['duffing']}
  4. Definitions and notations for \ref{['KG']}
  5. Main results for \ref{['KG']}
  6. Bibliographic comments
  7. Long-time behaviour of solutions.
  8. Ground states and related properties.
  9. Outline of the article
  10. Behaviour of solutions of the Duffing equation
  11. Proof of Proposition \ref{['prop_duffing_K']}
  12. Solutions with initial data in $\mathcal{K}^-$.
  13. Solutions with initial data in $\mathcal{K}^+$.
  14. Proof of Theorem \ref{['thm_duffing_N']}
  15. Step 1: the case $E(u_0,u_1) = \frac{1}{4}$.
  16. ...and 20 more sections

Key Result

Proposition 1

Let $(u_0, u_1) \in \mathcal{K}$ and $\gamma \geq 0$.

Figures (10)

  • Figure 1: The sets $\mathcal{K}^+$ (light gray), $\mathcal{K}^-$ (medium gray), and $\mathcal{N}$ (dark gray), separated by the curves $\{ E(u_0, u_1) = \frac{1}{4} \}$, given by $u_0 \mapsto \pm \frac{u_0^2 - 1}{\sqrt{2}}$ (in black).
  • Figure 2: Global existence vs blow-up for three different values of $\gamma$.
  • Figure 3: The sets $\mathcal{N}_1$ (light gray), $\mathcal{N}_2$ (medium gray), and $\mathcal{N}_3$ (dark gray).
  • Figure 4: For $(u_0,u_1) \in \mathcal{N}_1$, the solution blows up for all $\gamma \geq 0$.
  • Figure 5: For $(u_0,u_1) \in \mathcal{N}_2$, the solution blows up for small $\gamma$, and converges to zero for large $\gamma$. One can conjecture that there exists a unique intermediate value $\gamma_0$ such that the corresponding solution converges to $1$.
  • ...and 5 more figures

Theorems & Definitions (29)

  • Proposition 1
  • Theorem 2
  • Theorem 3
  • proof
  • Theorem 4
  • Proposition 5
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • ...and 19 more