数理解析研究所講究録
第 1779 巻 2012 年 88-90

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Global dynamics beyond the ground state energy
for nonlinear wave equations
Kenji Nakanishi(Kyoto University)

This is an introduction for the recent joint work with Wilhelm Schlag (Universityof Chicago) and Joachim Krieger (EPFL). We study global behavior of solutions forthe nonlinear Klein-Gordon equation (NLKG)$-\triangle u+u=|u|^{p-1}u$ ,$u(t, x)$ :$\mathbb{R}^{1+d}arrow \mathbb{R}$,$1+ \frac{4}{d}<p<1+\frac{4}{d-2}$(1),the nonlinear Schr\"odinger equation (NLS)$i\dot{v}-\Delta v=|v|^{p-1}v$,$v(t, x)$:$\mathbb{R}^{1+d}arrow \mathbb{C}$,$1+ \frac{4}{d}<p<1+\frac{4}{d-2}$(2),

and the nonlinear critical wave equation (NLW)$\ddot{w}-\triangle w=|w|^{4/(d-2)}w$,$w(t, x):\mathbb{R}^{1+d}arrow \mathbb{R}$,$d\geq 3$(3),

which have unstable ground states, i.e., the nontrivial stationary solution$u(t, x)=Q(x)$ ,$w(t, x)=W(x)$ ,(4)or the standing wave$v(t, x)=e^{-it\omega}Q(x)$$(\omega>0)$(5),

with the least energy and positive profiles. Here the conserved energy is defined
respectively by,$E(u)= \int_{\mathbb{R}^{d}}\frac{|\dot{u}|^{2}+|\nabla u|^{2}+|u|^{2}}{2}-\frac{|u|^{p+1}}{p+1}dx$(6)$E(v)= \int_{\mathbb{R}^{d}}\frac{|\nabla v|^{2}+\omega|v|^{2}}{2}-\frac{|v|^{p+1}}{p+1}dx$,.

$E(w)= \int_{\mathbb{R}^{d}}\frac{|\nabla w|^{2}}{2}-\frac{|w|^{p+1}}{p+1}dx$,
Hence the energy space is naturally defined by, in which we investigate the global dynamics.
orThe existence as well as the instability of the ground state is well known for
the above equations. $Q(x)$ decays exponentially as $|x|arrow\infty$ , while $W(x)$ decays
algebraically. The Lorentz or Galilei invariance of the equation generates a classof solitons with various traveling velocity from any stationary or standing wave
solution.
Besides those solitons, it is known that these equations have so-called scatteringsolutions, which are asymptotic to some solutions of the free equation (the equationin the energy space, and also blow-up solutions,
without the nonlinearity) as
diverges
energyor the energy concentrates in finite time (thefor which thenorm
latter happens only for the critical equation ). More precisely, it is characterized
by a finite maximal existence time of the unique local solution which is strongly$(u,\dot{u})\in H^{1}(\mathbb{R}^{d})\cross L^{2}(\mathbb{R}^{d}),$ $v\in H^{1}(\mathbb{R}^{d})$$(w,\dot{w})\in\dot{H}^{1}(\mathbb{R}^{d})\cross L^{2}(\mathbb{R}^{d})$$tarrow\infty$$w$

continuous in the energy space.
The goal of this study is to classify and predict global dynamics for all solutions or
initial data in the energy space, revealing the topological relation of solutions withdifferent evolutions. This is for now too ambitious, since there are too many possible

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scenarios and the global dynamics can be very complex. However, restricting theenergy at most slightly above the ground state, we can completely classify the globaldynamics, which shows in particular that even if the scattering, blowup and thesolitons are mixed together, the dynamics can stay reasonably ordered, thanks to
the global dispersion of the equations.
More precisely, our main results state that all initial data in the energy spacewith energy at most slightly above the ground state are split into 9 sets, according
to the global behavior of the solution, which include scattering, solitons and blowup, as well as transitions among them from $t<0$ to $t>0$ . The classification is
given in terms of center-stable and center-unstable manifolds of the ground state,constructed globally under the energy restriction, as the hypersurfaces separating
scattering and blowup solutions for $t>0$ and $t<0$ , respectively. For simplicity,
the full statement is given below in the model case: radial, cubic and $3D$ NLKG [3].
The other cases [5, 2, 4, 1, 6] can be understood as extension of it.

Theorem 1. Let $d=p=3$ for (NLKG). For each, let
mal subsets ofsuch that for each initial data
the corresponding solution of NLKG satisfies$\sigma=\pm$$S_{\sigma},$$\mathcal{B}_{\sigma},$$:=H^{1}(\mathbb{R}^{3})\cross L^{2}(\mathbb{R}^{3})$$\mathcal{H}$$\mathcal{T}_{\sigma}$be the maxi-$(u(O),\dot{u}(0))\in \mathcal{H}$,$u$scatters as$(u(O),\dot{u}(0))\in B_{\sigma}\Rightarrow u$
blows up in
scatters as$(u(O), (0))\in S_{\sigma}\Rightarrow u$$\sigma tarrow\infty$$\sigma t>0$$(u(O),\dot{u}(0))\in \mathcal{T}_{\sigma}\Rightarrow u\mp Q$

For each subset$X\subset \mathcal{H}$and$\epsilon>0$, let$X^{\epsilon}$$\epsilon>0$$g$(7).

be the restriction.

(8)such that the following holds. We have$\mathcal{H}^{\epsilon}=S_{+}^{\epsilon}\cup \mathcal{B}_{+}^{\epsilon}\cup \mathcal{T}_{+}^{\epsilon}=S_{-}^{\epsilon}\cup \mathcal{B}^{\underline{\epsilon}}\cup\underline{\mathcal{T}^{\epsilon}}$and each of the,$\sigma tarrow\infty$$X^{\epsilon}=\{\varphi\in X|E(\varphi)<E(Q)+\epsilon^{2}\}$

Then there exists,(disjoint union),(9)intersections$S_{+}^{\epsilon}\cap S_{-}^{\epsilon}$$\mathcal{B}_{+}^{\epsilon}\cap S^{\underline{\epsilon}}$$\mathcal{T}_{+}^{\epsilon}\cap S_{-}^{\epsilon}$,,,,,,$S_{+}^{\epsilon}\cap \mathcal{B}_{-}^{\epsilon}$$\mathcal{B}_{+}^{\epsilon}\cap \mathcal{B}_{-}^{\epsilon}$$\mathcal{T}_{+}^{\epsilon}\cap \mathcal{B}_{-}^{\epsilon}$,$S_{+}^{\epsilon}\cap\underline{\mathcal{T}^{\epsilon}}$$\mathcal{B}_{+}^{\epsilon}\cap\underline{\mathcal{T}^{\epsilon}}$

,

(10)

$\mathcal{T}_{+}^{\epsilon}\cap \mathcal{T}_{-}^{\epsilon}$

contains infinitely many orbits.
andare unbounded connected open sets,are unbounded smooth manifold of codimension 1 with two connected components,
andis a smooth manifold of codimension 2 with two components within
distance $from\pm Q$ .
$S_{\pm}^{\epsilon}$$\mathcal{B}_{\pm}^{\epsilon}$$\mathcal{T}_{\pm}^{\epsilon}$$\mathcal{T}_{+}^{\epsilon}\cap \mathcal{T}^{\underline{\epsilon}}$$O(\epsilon)$

The key ingredient in the proof is what we call the one-pass theorem, which
precludes orbits departing from and returning to a small neighborhood of the groundstate. Once it is established, the analysis is essentially decomposed into a scattering
region, a blowup region and the small neighborhood of the ground state. In each ofthem, some modffication of the preceding works suffices.

Theorem 2. Let $d=p=3$ and. Then there are small,-neighborhood $Uof\pm Q$ , a continuous function 6:
such that$\epsilon>0$$\mathcal{H}=H^{1}(\mathbb{R}^{3})\cross L^{2}(\mathbb{R}^{3})$$O(\epsilon)$$\mathcal{H}^{\epsilon}\backslash Uarrow\{\pm 1\}$

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the following hold. For any solution
existence interval, and$u$of NLKG in$\mathcal{H}^{\epsilon}$, let$I(u)\subset \mathbb{R}$be the maximal$I_{S}(u)=\{t\in I(u)|6((u(t),\dot{u}(t))=+1\}$ ,(11)i15 $((u(t),\dot{u}(t))=-1\}$ ,$I_{\mathcal{T}}(u)=\{t\in I(u)|(u(t),\dot{u}(t))\in U\}$ .
together have at most two connected
andThenis connected, soconsists of boundedcomponents.
consists of unbounded intervals, while$(u(O),\dot{u}(0))\inx_{+}$
such that, we haveintervals. For each $X=S,$iff$(T, \infty)\cap I(u)\subset I_{X}(u)$ . For those
close to $(\pm Q, 0)$ , we have
$I_{B}(u)=\{t\in I(u)|$$I_{B}(u)$$I_{S}(u)$$I_{\mathcal{T}}(u)$$($$I_{S}(u)$$I_{\mathcal{B}}(u)$$\mathcal{B},$$\exists T\in I(u)$$\mathcal{T}$$(\varphi, \psi)\in \mathcal{H}$$\mathfrak{S}((\varphi, \psi))=$sign$\{Q\mp\varphi|\rho\}$,(12)

where is the ground state of the lineartzed $opemtor-\triangle+1-3Q^{2}$ , and foraway from $(\pm Q, 0)$ , we have$\rho$

6
Moreover,$\mathfrak{S}$$((\varphi, \psi))=$sign$\int_{\mathbb{R}^{3}}[|\nabla\varphi|^{2}+|\varphi|^{2}-|\varphi|^{4}]dx$.

$(\varphi, \psi)$(13)

is uniquely determined by the above two formulas.

The one-pass theorem is proved by combinlng the hyperbolic dynamics of thelinearized equation near but slightly away from the ground state, and the variational
analysis far from the ground state, together with a virial identity localized in spacetime for an almost homoclinic orbit.
Despite the lengthy statements, the proofs are not technically complicated at all,which are based on numerous classical and recent ideas. The interested reader isinvited to look first into the above simplest setting, either in the paper [3] or in thebook [7], where one needs no estimate or computation which lasts for several lines.

REFERENCES[1] J. Krieger, K. Nakanishi and W. Schlag, Global dynamics away from the ground state for theenergy-cretical nonlinear wave equation. arXiv:1010.3799, to appear in Amer. J. Math.
[2] J. Krieger, K. Nakanishi and W. Schlag, Global dynamics above the ground state energy for theone-dimensional NLKG equation. arXiv:lO11.1776, to appear in Math. Z.
[3] K. Nakanishi and W. Schlag, Global dynamics above the ground state energy for the focusing
nonlinear Klein-Gordon equation. J. Differential Equations 250 (2011), no. 5, 2299-2333.
[4] K. Nakanishi and W. Schlag, Global dynamics above the ground state energy for the cubic NLS
equation in 3D. arXiv:1007.4025, to appear in Calc. Var. Partial Differential Equations.
[5] K. Nakanishi and W. Schlag, Global dynamics above the ground state for the nonlinear KleinGordon equation without a radial assumption. arXiv:lO11.0132, to appear in Arch. Ration.
Mech. Anal.
[6] K. Nakanishi and W. Schlag, Invariant manifolds around soliton manifolds for the nonlinear
Klein-Gordon equation. arXiv:1102.5583.
[7] K. Nakanishi and W. Schlag, Invariant manifolds and dispersive Hamiltonian evolution equations, Z\"urich Lectures in Advanced Mathematics, EMS, 2011.