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Nakanishi, K.; Schlag, W.

Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation. (English) Zbl 1213.35307

J. Differ. Equations 250, No. 5, 2299-2333 (2011).
This paper deals with the focusing, cubic, nonlinear Klein-Gordon equation in \(\mathbb{R}^3_x\) with large radial data in the energy space: \[ \ddot u-\Delta u+ u= u^3,\;u= u)t,x);\;\vec u(t)= (u(t),\dot u(t))\in{\mathcal H}= H^1_{\text{rad}}\times L^2_{\text{rad}}.\tag{1} \] As it is well-known, (1) possesses a unique positive radial stationary solution \(Q(x)\), called the ground state and having the least energy \(E(Q)> 0\) among the static solutions.
The authors propose a complete description of the evolution of \(u(t,x)\) in that case: \(E(\vec u)< E(Q)+ \varepsilon^2\), \(0<\varepsilon\ll 1\). More precisely, they verify the following trichotomy in a small neighborhood of \(Q\): On one side of a center-stable manifold one has finite time blow-up for \(t\geq 0\), on the other side scattering to zero, and on the manifold itself one has scattering to \(Q\), both as \(t\to+\infty\). The above-mentioned class of data \(\vec u(0)\) with energy at most slightly above that of \(A\), is divided into 9 disjoint non-empty sets each displaying different asymptotic behaviour as \(t\to\pm\infty\), which includes solutions blowing up in one time direction and scattering to \(0\) on the other. For example, case (7) asserts that the solution \(u\) is trapped by \(\pm Q\) for \(t\to+\infty\) (i.e. J\(u\) in an \(\varepsilon\)-neighbourhood of \(\pm Q\) after some time) and possesses finite time blow-up in \(t< 0\).
The proof relies on an “one pass” theorem that excludes the existence of homoclinic orbits between \(Q\) (as well \(-Q\)) and heteroclinic orbits connecting \(Q\) with \(-Q\).
Reviewer: Petar Popivanov (Sofia)

Cited in 6 Reviews
Cited in 29 Documents

MSC:

35L70 Second-order nonlinear hyperbolic equations
35Q55 NLS equations (nonlinear Schrödinger equations)

Keywords:

nonlinear Klein-Gordon equation; ground state; stable manifold; scattering theory; blow up; unstable manifold
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\textit{K. Nakanishi} and \textit{W. Schlag}, J. Differ. Equations 250, No. 5, 2299--2333 (2011; Zbl 1213.35307)
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