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

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