Center-stable manifold of the ground state in the energy space for the critical wave equation

2013/03/11 by Joachim Krieger, Krieger, Joachim, Kenji Nakanishi +3 · 2 citations
Mathematics · Physics and Astronomy · #35B40 #35L70 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.1303.2400

openalex publication_date 2013/03/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We construct a center-stable manifold of the ground state solitons in the energy space for the critical wave equation without imposing any symmetry, as the dynamical threshold between scattering and blow-up, and also as a collection of solutions which stay close to the ground states. Up to energy slightly above the ground state, this completes the 9-set classification of the global dynamics in our previous paper. We can also extend the manifold to arbitrary energy size by adding large radiation. The manifold contains all the solutions scattering to the ground state solitons, and also some of those blowing up in finite time by concentration of the ground states.

Citations

• Saddle points and instability of nonlinear hyperbolic equations

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• Blow-up, decay, and convergence to equilibrium for focusing damped cubic Klein-Gordon and Duffing equations
• Dynamics of 3D focusing, energy-critical wave equation with radial data

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