Global dynamics beyond the ground energy for the focusing nonlinear

Klein-Gordon equation

Kenji Nakanishi (Kyoto Univ.)

Joint work withWilhelm Schlag (Univ. of Chicago)

Ultimate Goal: Classify & Predict global dynamics of all solutions for nonlinear dispersive equations. Our result ： for the simplest model （ＮＬＫＧ） 　　 the nonlinear Klein-Gordonequation, we have classification into 9 sets,including “scattering”, “soliton”, and “blowup”,and transitions between them, (hence 3×3)under a restriction on the total energy.　　　　　 (No smallness or symmetry assumption)Moreover, we can partially predict the dynamics.

The conserved energy is (focusing nonlinearity)

the static energy is

NLKG may be regarded as a Hamiltonian system for

The norm (free energy) is denoted by

conserved for the free solution KG(u) = 0.

Energy space

Typical 3 nonlinear dispersive waves1. Scattering: global & asymptotic to a free sol.

2. Soliton: traveling wave with localized shape ex.) Ground state:　 minimal energy J(Q)>0 among stat. solutions. Lorentz transformed into traveling waves:

: speed 3. Blowup: free energy diverges in finite time.

Since free sol. v decays in , u decays too, by Sobolev (2< p ≤6)

traveling waves have non-zero constant norms for any p

,...),()( 21 xstcxQxQ

　　 Scaling landscape of the static energy J() Ground state Q is a saddle pt. of J, unstable in NLKG. E < J(Q) is divided by the sign of scaling derivative K, which is invariant. They are connected in E ≥ J(Q) .

connected

separated

Scaling derivative of static energy

global

blowup

　 Derivative of J by 　 -preserving scaling (s=0 or 2)

is , and Q is the minimizer of

For E(u) < J(Q) either [u = 0 or K > 0] or K < 0Payne-Sattinger (‘75): (bounded domains)

Kenig-Merle (‘06): -wave/NLS ( -critical) scatter Ibrahim-Masmoudi-N. (‘09): NLKG, general power

What if E(u) ≥ J(Q) ?E(u)=J(Q): Duyckaerts-Merle (‘08): -wave/NLS Only 3 new solutions besides Kenig-Merle : Q, and 　　 (Unique modulo the symmetry of the equation)Near Q: Bates-Jones (‘89):Under the radial restriction, Q M : center-stable mfd. codim=1 H (radial)Schlag (‘04), Beceanu (‘09) NLS, w/o radial restriction {soliton family} M : center-stab. mfd. codim=1, sol. starting on M scatters to {soliton family}

)(:),(:)(

tWtWtQW

upblowscatter

invariant for the forward evolution by NLKG, tangent to the center-stable subspace at Q

Linearization at Q is

(for smaller power, (0,1] also contains eigenvalues)

ScatteringBlowup

Main result (radial case for simplicity)

Define initial data sets by

“trapped by Q” (as 　 　　　 ) means

similarly, 　 are defined for negative time. Restriction to are denoted by etc. Then

All the 9 sets are infinite, even modulo symmetry. are open & connected, separated by . have two components, respectively with Q. 　 is within distance O() from Q. In particular, no heteroclinic orbit between Q. is an unbounded manifold with codim=1, separating locally and globally into and . is a bounded manifold with codim=2.The division into scattering/blowup is given by the center-(un)stable manifolds (not by sign(K))! These can be proved without the spectral gap in (0,1], and for any dimension d, power 1+4/d < p < 1+4/(d-2). By using the fine spectral property, we can further show that

S: ScatteringB:Blow-upT: Trapped by Q

Phase space .tubes: (un)stable mfd. dim=1. 　 center mdf. codim=2.meshed surfaces: center-(un)stable mfd. codim=1. inner surface: E<E(Q). outer surface: E<E(Q)+

Scattering (∞)

Blow-up

(∞)

Q

Center-stablemanifold

Center-unstablemanifold

Scattering (−∞)Blowup(∞)

Blowup(−∞)Scattering(∞)

Key: nonexistence of “almost homoclinic orbit” One-pass Theorem : Transition from “scattering” to “blowup” happens only by passing near Q, which is possible at most once for every solution. Proved by using the virial identity localized into 2 cones, and the finite propagation.

Q

One-pass theoremsmall >0, B: O() –neighbor hood of Q, continuous functional, s.t. for every solution u in , let

Then is an interval, consists of at most 2 unbdd. intervals, at most 2 bdd. intervals scatters in t > 0 ⇔ S=+1 for large t. blows up in t > 0 ⇔ S= – 1 for large t. trapped in t > 0 ⇔ staying in B for large t.

　 Moreover, S is opposite to the unstable mode of u-Q near Q, while it equals sign(K) away from Q.

　 Non-radial caseLorentz transform changes E(u) and the (conserved) momentum P(u)by On P = 0, we have the same results as in the radial case, which is Lorentz transformed intothe set where the dynamics on is given by “scattering to the family of the traveling waves”:

where is the traveling wave generated from Q, having the relativistic momentum p and position q.

After getting one-pass: the solution stays in either S = +1 (scattering), S = － 1 (blowup), or B (near

Q), then the remaining argument is similar to Kenig-Merle, Payne-Sattinger, or Schlag-Beceanu.

Q Btrapped

<< 1hyperbolicexp. growth

variationalsign definite

energydistance

t = S: starting near Qt = T: returning close to Q

Rough idea of the proofTo prove One-pass theorem: 1. Hyperbolic dynamics near Q: dominated by ⇒ convex & exponential dynamics of the energy distance from Q2. Variational estimate away from Q : mountain-

pass ⇒ lower bound on |K| Patching 1 & 2 monotonicity in time average of ⇒ the localized virial for “almost homoclinic orbits”.

This requires a lot of spectral property of L for global estimates, not easy to extend to general case (e.g. d & p). Alternatively, we can extend Bates-Jones to the non-radial case with traveling waves. The main difficulty in their way (Hadamard) is: Spatial translation is not Lipschitz in H. We overcome it by using a nonlinear quasi-distance with the same topology where the translation is Lipschitz. The latter applies to radial excited states too.

Further results and Open problemsThe above results are extended to: • NLS (3D-cubic) in the radial case. • -nonlinear wave in the radial case, except for the dynamics on the trapped sets (joint with Krieger and Schlag) Major open questions• What if E(u) is much bigger than J(Q), at least below the energy of the first excited state ? • What if the nonlinear power p < 1+4/d ?• What if the ground state is (orbitally) stable?• Blow-up profiles ?• Topological properties of the 9 sets ?