Scattering theory for energy-supercritical Klein-Gordon equation

0.28 MB·English

by  Changxing Miao

SCATTERING THEORY FOR ENERGY-SUPERCRITICAL
KLEIN-GORDON EQUATION
CHANGXINGMIAO ANDJIQIANGZHENG
3
1 Abstract. Inthispaper, weconsiderthequestionof theglobal well-posedness and
0 scatteringforthecubicKlein-Gordonequationu −∆u+u+|u|2u=0indimension
tt
2 d > 5. We show that if the solution u is apriorily bounded in the critical Sobolev
n space, that is, (u,ut) ∈ L∞t (I;Hxsc(Rd)×Hxsc−1(Rd)) with sc := d2 −1 > 1, then
a u is global and scatters. The impetus to consider this problem stems from a series
J of recent works for the energy-supercritical nonlinear wave equation and nonlinear

1 Schr¨odingerequation. However,thescalinginvarianceisbrokenintheKlein-Gordon
2 equation. Wewillutilizetheconcentrationcompactnessideastoshowthattheproof
oftheglobalwell-posednessandscatteringisreducedtodisprovetheexistenceofthe

] scenario: soliton-like solutions. And such solutions are precluded by making use of
P
the Morawetz inequality, finite speed of propagation and concentration of potential
A
energy.
.
h Key Words: Klein-Gordon equation; scattering theory; Strichartz
at estimate; Energy supercritical; concentration compactness
m AMS Classification: 35P25,35B40, 35Q40.
[
1. Introduction
2
v This paper is devoted to the study of the Cauchy problem of the cubic Klein-Gordon
6
equation
6
6
u¨−∆u+u+f(u)= 0, (t,x) ∈ R×Rd, d > 5,
4
(1.1)
1. ( u(0,x), ut(0,x) = u0(x), u1(x) ∈ Hsc(Rd)×Hsc−1(Rd),
1
2 where f(u) =(cid:0) |u|2u, u is a re(cid:1)al-v(cid:0)alued function(cid:1) defined in R1+d, the dot denotes the
1 time derivative, ∆ is the Laplacian in Rd, s := d −1.
: c 2
v Formally, the solution u of (1.1) conserves the energy
i
X 1 1
E(u(t),u˙(t)) = u˙(t,x) 2+ ∇u(t,x) 2+ u(t,x) 2 dx+ |u|4dx
ar ≡E2(ZuRd,(cid:16)u(cid:12)). (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:17) 4 ZRd
0 (cid:12)1 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
The class of solutions to wave equation
u¨−∆u+|u|2u = 0
is left invariant by the scaling
(1.2) u(t,x) 7→ λ u(λ t,λ x), ∀ λ > 0.
Moreover, it leaves the Sobolev norm H˙sc(Rd) with s = d −1 invariant. Since s > 1,
x c 2 c
it is called the energy-supercritical.
1
2 CHANGXINGMIAOANDJIQIANGZHENG
The scattering theory for the Klein-Gordon equation with f(u)= µ|u|p−1u has been
intensively studied in [3,4,12,14,26,27]. For µ = 1 and
1, 3 6 d6 9;
4 4
(1.3) 1+ < p < 1+γd , γd = d
d d−2  , d > 10.
 d+1
Brenner [4] established the scattering results inthe energy space H1(Rd) × L2(Rd),
x x
whichdoesnotcontainallsubcriticalcasesford > 10. Thereafter,GinibreandVelo[12]
exploited the Birman-Solomjak space ℓm(Lq,I,B) in [2] and the delicate estimates to
improve the results in [4], which covered all subcritical cases. Finally K. Nakanishi [26]
obtained the scattering results for the critical case (p = 1+ 4 ) by the strategy of
d−2
induction on energy [9] and a new Morawetz-type estimate. And recently, S. Ibrahim,
N. Masmoudi and K. Nakanishi [14,15] utilized the concentration compactness ideas
to give the scattering threshold for the focusing (µ = −1) nonlinear Klein-Gordon
equation. Their method also works for the defocusing case.
In this paper, we consider the cubic Klein-Gordon equation in dimension d > 5,
which is the energy-supercritical case. Such results have been recently established for
many other equations including the nonlinear wave equation (NLW) and the nonlinear
Schro¨dinger equation (NLS), since Kenig-Merle [18] on NLW for radial solutions in R3.
We also refer to [5–7,10,19–22].
To be more precise, let us recall the results for the energy-supercritical nonlinear
wave equation
4
u¨−∆u+|u|pu= 0, (t,x) ∈R×Rd, p > .
d−2
Indimensionthree, KenigandMerle[18]proved thatiftheradialsolution uisapriorily
bounded in the critical Sobolev space, that is, (u,u ) ∈ L∞(I;H˙sc(Rd)×H˙sc−1(Rd))
t t x x
with s := d − 2 > 1, then u is global and scatters. In [19], they also considered the
c 2 p
radial solutions in odd dimensions. Later, Killip and Visan [21] showed the result in R3
for thenon-radialsolutions bymaking useof Huygens principalandsocalled “localized
double Duhamel trick”. Further, they [22] proved the radial solution in all dimensions
in some ranges of p. Thereafter, Bulut [5–7] proved the results in dimensions d> 5 for
the cubic nonlinearity (i.e. p = 2). Recently, Duyckaerts, Kenig and Merle [10] obtain
such result for the focusing wave equation with radial solution in three dimension.
Theirproof relies on thecompactness/rigidity method,pointwise estimates on compact
solutions obtained in [18], and channels of energy arguments used by the authors in
previous works [11] on the energy-critical equation.
Before stating the main result, we introduce some background materials.
Definition 1.1 (solution). A function u : I ×Rd → R on a nonempty time interval
I containing zero is a strong solution to (1.1) if (u,u )∈ C0(J;Hsc(Rd)×Hsc−1(Rd))
t t x x
and u ∈ [W](J) (defined in (1.8)) for any compact interval J ⊂ I and for each t ∈ I,
it obeys the Duhamel formula:
u(t) u (x) t 0
0
(1.4) = V (t) − V (t−s) ds,
0 0
u˙(t) u (x) f(u(s))
(cid:18) (cid:19) (cid:18) 1 (cid:19) Z0 (cid:18) (cid:19)
SCATTERING THEORY FOR ENERGY-SUPERCRITICAL KLEIN-GORDON EQUATION 3
where
K˙(t),K(t) sin(tω)
1/2
V (t) = , K(t) = , ω = 1−∆ .
0 K¨(t),K˙(t) ω
(cid:18) (cid:19)
We refer to the interval I as the lifespan of u. We say that(cid:0)u is a(cid:1)maximal-lifespan
solution if the solution cannot be extended to any strictly larger interval. We say that
u is a global solution if I = R.
The solution lies in the space [W](I) locally in time is natural since by Strichartz
estimate, the linear flow always lies in this space. Also, if a solution u to (1.1) is global,
with kukW(R) < +∞, then it scatters in both time directions in the sense that there
exist solutions v of the free Klein-Gordon equation
±
(1.5) v¨−∆v+v = 0
with (v (0),v˙ (0)) ∈ Hsc(Rd)×Hsc−1(Rd) such that
± ± x x
(1.6) u(t),u˙(t) − v (t),v˙ (t) −→ 0, as t −→ ±∞.
± ± Hxsc×Hxsc−1
(cid:13) (cid:13)
In view of t(cid:13)h(cid:0)is, we defi(cid:1)ne (cid:0) (cid:1)(cid:13)
(cid:13) (cid:13)
(1.7) S (u) = kuk
I [W](I)
as the scattering size of u, where
2(d+1) d−3
(1.8) [W](I) = L d−1 I;B 2 (Rd) .
t 2(d+1),2
d−1
(cid:0) (cid:1)
Closely associated with the notion of scattering is the notion of blowup:
Definition 1.2 (Blowup). Let u : I × Rd → C be a maximal-lifespan solution to
(1.1). If there exists a time t ∈ I such that S (u) = +∞, then we say that the
0 [t0,supI)
solution u blows up forward in time. Similarly, if there exists a time t ∈ I such that
0
S (u) = +∞, then we say that u(t,x) blows up backward in time.
(inf I,t0]
Now we state our main result.
Theorem 1.1. Assume that d > 5, and s := d − 1. Let u : I × Rd → R be a
c 2
maximal-lifespan solution to (1.1) such that
(1.9) (u,u ) < +∞.
t L∞t (I;Hxsc(Rd)×Hxsc−1(Rd))
Then the solution u is g(cid:13)lobal an(cid:13)d scatters.
(cid:13) (cid:13)
The outline of the proof of Theorem 1.1: For any 06 E < +∞, we define
0
L(E ) := sup S (u) : u: I ×Rd → R such that sup (u,u ) 2 6 E ,
0 I t∈I t Hxsc×Hxsc−1 0
n (cid:13) (cid:13) o
where the supremum is taken over all solutions u : I ×(cid:13)Rd →(cid:13)R to (1.1) satisfying
2
(u,u ) 6 E . Thus, L : [0,+∞) → [0,+∞) is a non-decreasing function.
t Hsc×Hsc−1 0
Moreover, from the small data theory, see Theorem 2.1, we know that
(cid:13) (cid:13)
(cid:13) (cid:13)
1
L(E ) . E2 for E 6 η2,
0 0 0 0
where η = η(d) is the threshold from the small data theory.
0
4 CHANGXINGMIAOANDJIQIANGZHENG
From the stability theory (see Theorem 2.5), we see that L is continuous. Therefore,
there must exist a unique critical E ∈ (0,+∞] such that L(E ) < +∞ for E < E
c 0 0 c
and L(E ) = +∞ for E > E . In particular, if u : I ×Rd → R is a maximal-lifespan
0 0 c
2
solution to (1.1) such that sup (u,u ) < E , then u is global and moreover,
t Hsc×Hsc−1 c
t∈I
(cid:13) (cid:13)
SR(u)(cid:13)6 L (u(cid:13),ut) 2L∞(R;Hsc×Hsc−1) .
t
The proof of Theorem 1.1 is equiva(cid:0)l(cid:13)ent to s(cid:13)how E = +∞. W(cid:1)e argue by contradiction.
(cid:13) (cid:13) c
We show that if E < +∞, then there exists a nonlinear global solution of (1.1) with
c
L∞(R;Hsc ×Hsc−1)-norm be exactly E . Moreover, this solution satisfies some strong
t x x c
compactness properties. This is completed in Section 4 where we utilize the profile
decomposition that was established in Ibrahim, Masmoudi and Nakanishi [14], and a
strategy introduced by Kenig and Merle [17]. Finally, we utilize the finiteness of the
energytoshowthatthesolutionsobtainedinSection4arenotpossible. Moreprecisely,
by Morawetz inequality [4,23,25]
T |u(t,x)|4
(1.10) dxdt . E(u,u ),
t
Z0 ZRd |x|
we know that the left-hand side of (1.10) is bounded by the energy for any T > 0.
On the other hand, notice that by finite speed of propagation and concentration of
potential energy, the left-hand side of (1.10) should grow logarithmical in time T. This
gives a contradiction by choosing T sufficiently large.
The paper is organized as follows. In Section 2, we deal with the local theory for
the equation (1.1). In Section 3, we give the linear and nonlinear profile decomposition
and show some properties of the profile. Thereafter, we extract a critical solution in
Section 4. Finally in Section 5, we preclude the critical solution, which completes the
proof of Theorem 1.1.
Weconcludetheintroductionbygivingsomenotationswhichwillbeusedthroughout
this paper. We always assume the spatial dimension d > 5 and f(u) = |u|2u. For any
r,1 6 r 6 ∞, we denote by k·k the norm in Lr = Lr(Rd) and by r′ the conjugate
r
exponent defined by 1+ 1 = 1. For any s ∈R, we denote by Hs(Rd) the usualSobolev
r r′
space. Let ψ ∈ S(Rd) be such that supp ψ ⊆ ξ : 1 6 |ξ| 6 2 and ψ(2−jξ) = 1
2 j∈Z
for ξ 6= 0. Define ψ by ψ = 1− ψ(2−jξ). Thus supp ψ ⊆ ξ : |ξ| 6 2 and
0 0 j>1 (cid:8) (cid:9) 0 P
b b
ψ = 1 for |ξ| 6 1. We denote by ∆ and P the convolution operators whose symbols
0 Pj 0 (cid:8) (cid:9)
are respectively given bybψ(ξ/2j) and ψ (bξ). For s ∈ R,1 6 rb6 ∞, the Besov spaces
0
Bbs (Rd) and B˙s (Rd) are defined by
r,2 r,2
b b
Bs (Rd)= u∈ S′(Rd),kP uk2 + 2jsk∆ uk 2 < ∞
r,2 0 Lr j Lr lj2∈N
(cid:26) (cid:27)
(cid:13) (cid:13)
B˙s (Rd)= u∈ S′(Rd), 2jsk∆ uk(cid:13) 2 < ∞ (cid:13).
r,2 h j Lr lj2∈Z
(cid:26) (cid:27)
For details ofBesov space, wereferto[(cid:13)(cid:13)1]. For any in(cid:13)(cid:13)tervalI ⊂ Randany Banach space
X we denote by C(I;X) the space of strongly continuous functions from I to X and by
Lq(I;X)thespaceofstronglymeasurablefunctionsfromI toX withku(·);Xk ∈Lq(I).
We denote by h·,·i the scalar product in L2.
SCATTERING THEORY FOR ENERGY-SUPERCRITICAL KLEIN-GORDON EQUATION 5
2. Preliminaries
2.1. Strichartz estimate and local theory. In this section, we consider the Cauchy
problem for the equation (1.1)
u¨−∆u+u+f(u)= 0,
(2.1)
u(0) = u , u˙(0) = u .
( 0 1
The integral equation for the Cauchy problem (2.1) can be written as
t
(2.2) u(t) = K˙(t)u +K(t)u − K(t−s)f(u(s))ds,
0 1
Z0
or
u(t) u (x) t 0
0
(2.3) = V (t) − V (t−s) ds,
0 0
u˙(t) u (x) f(u(s))
(cid:18) (cid:19) (cid:18) 1 (cid:19) Z0 (cid:18) (cid:19)
where
sin(tω) K˙(t),K(t)
1/2
K(t)= , V (t) = , ω = 1−∆ .
ω 0 K¨(t),K˙(t)
(cid:18) (cid:19)
Let U(t)= eitω, then (cid:0) (cid:1)
U(t)+U(−t) U(t)−U(−t)
K˙(t)= , K(t) = .
2 2iω
Now we recall the following dispersive estimate for the operator U(t) = eitω.
Lemma 2.1 ( [4,12]). Let 26 r 6 ∞ and 0 6 θ 6 1. Then
eiωtf 6 µ(t) f ,
Br−,2(d+1+θ)(21−1r)/2 Br(d′,+21+θ)(21−r1)/2
(cid:13) (cid:13) (cid:13) (cid:13)
where (cid:13) (cid:13) (cid:13) (cid:13)
µ(t) =Cmin |t|−(d−1−θ)(12−r1)+,|t|−(d−1+θ)(12−1r) .
(cid:26) (cid:27)
According to the above lemma, the abstract duality and interpolation argument(see
[13], [16]), we have the following Strichartz estimates.
Lemma 2.2 ([4,12,23,24]). Let 0 6 θ 6 1, ρ ∈ R, 2 6 q ,r 6 +∞, i= 1,2. Assume
i i i i
that (θ ,d,q ,r )6= (0,3,2,+∞) satisfy the following admissible conditions
i i i
2 1 1
0 6 6 min (d−1+θ ) − ,1 , i = 1,2
i
q 2 r
i i
 n (cid:16) (cid:17) o
1 1 1
(2.4)  ρ1+(d+θ1) − − = µ,
 2 r q
 1 1
 (cid:16) (cid:17)
1 1 1
ρ +(d+θ ) − − = 1−µ.
2 2
 2 r2 q2
 (cid:16) (cid:17)
Then, for g ∈ Hµ(Rd), we have
x
(2.5) U(·)g 6 Ckgk ;
Lq1 R;Bρ1 Hµ
r1,2
(2.6) (cid:13)(cid:13)KR∗f(cid:13)(cid:13)Lq1(cid:0)I;Brρ11,2(cid:1) 6 C f Lq2′ I;Br−2′ρ,22 .
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:0) (cid:1) (cid:13) (cid:13) (cid:0) (cid:1)
6 CHANGXINGMIAOANDJIQIANGZHENG
where the subscript R stands for retarded, and
t
K ∗f = K(t−s)f(u(s))ds.
R
Z0
Now it is useful to define several spaces and give estimates of the nonlinearities in
terms of these spaces. Define
ST(I) = [W](I),
where
2(d+1) d−3
[W](I) = L d−1 I;B 2 (Rd) .
t 2(d+1),2
d−1
In addition to the ST-norm, we also need t(cid:0)he correspondin(cid:1)g dual norm
2(d+1) d−3
[W]∗(I) = L d+3 I;B 2 (Rd) .
t 2(d+1),2
d+3
(cid:0) (cid:1)
Then we have by Strichartz estimate
u + (u,u )
[W](I) t L∞t (I;Hxsc×Hxsc−1)
(2.7) 6C(cid:13) (cid:13)(u ,u ) (cid:13) (cid:13) +C f(u) ,
(cid:13) (cid:13) 0 1 H(cid:13)xsc×Hxsc(cid:13)−1 [W]∗(I)⊕L1t(I;B2s,c1−1(Rd))
(cid:13) (cid:13) (cid:13) (cid:13)
where the time interval I contains zero.
(cid:13) (cid:13) (cid:13) (cid:13)
Lemma 2.3 (Product rule [8,28]). Let s > 0, and 1 < r,p ,q < +∞ be such that
j j
1 = 1 + 1 (i = 1,2). Then, we have
r pi qi
|∇|s(fg) Lrx(Rd) . kfkLpx1(Rd) |∇|sg Lqx1(Rd)+ |∇|sf Lpx2(Rd)kgkLqx2(Rd).
As a(cid:13)direct con(cid:13)sequence, we have t(cid:13)he foll(cid:13)owing nonl(cid:13)inear e(cid:13)stimate.
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
Lemma 2.4 (Nonlinear estimate). Let I be a time slab, one has
u2v
[W]∗(I)
(cid:13) 1+(cid:13) 2 d−3 2 d−3 4 2(d−3)
(2.8) .(cid:13)u (cid:13)d−1 u d−1 v d−1 v d−1 + v u d−1 u d−1 .
[W](I) L∞t H˙xsc [W](I) L∞t H˙xsc [W](I) [W](I) L∞t H˙xsc
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
Proof. ItfollowsfromtheaboveproductruleandSobolevembedding: Bs (Rd) ⊂
p,min{p,2}
Fs (Rd)= Ws,p(Rd) ⊂ Bs (Rd) that
p,2 x p,max{p,2}
(2.9) u2v . u u v + v u 2 .
[W]∗(I) [W](I) Ld+1 Ld+1 [W](I) Ld+1
t,x t,x t,x
Using H¨old(cid:13)er’s(cid:13)inequality(cid:13)an(cid:13)d Sobo(cid:13)lev(cid:13)emb(cid:13)ed(cid:13)ding, we(cid:13)ob(cid:13)tain (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
2 d−3
u . u d−1 u d−1
Ld+1 2(d+1) 2d(d+1) L∞Ld
(cid:13) (cid:13) t,x (cid:13) (cid:13)Lt d−1 Lx d+3 (cid:13) (cid:13) t x
(cid:13) (cid:13) (cid:13) (cid:13) 2 d−3 (cid:13) (cid:13)
. u d−1 u d−1 .
[W](I) L∞H˙sc
t
(cid:13) (cid:13) (cid:13) (cid:13)
Plugging this into (2.9), we get (2.8). (cid:3)
(cid:13) (cid:13) (cid:13) (cid:13)
SCATTERING THEORY FOR ENERGY-SUPERCRITICAL KLEIN-GORDON EQUATION 7
We can now state the local well-posedness for (1.1) with large initial data and small
data scattering in the space Hsc(Rd)×Hsc−1(Rd), which is the first step to obtain the
global time-space estimate and then lead to the scattering.
Theorem 2.1 (Local wellposedness). Assume (u ,u ) ∈ Hsc(Rd)×Hsc−1(Rd). There
0 1 x x
exists a small constant δ = δ(E) such that if k(u0,u1)kHsc×Hsc−1 6 E and I is an time
interval containing zero such that
(2.10) K˙(t)u +K(t)u 6 δ,
0 1 [W](I)
then there exists a unique stro(cid:13)ng solution u to ((cid:13)1.1) in I×Rd, with u∈ C(I;Hsc(Rd))∩
(cid:13) (cid:13) x
C1(I;Hsc−1(Rd)) and
x
(2.11) kuk[W](I) 6 2δ, k(u,ut)kL∞(I;Hsc×Hsc−1) 6 2CE,
t
where C is the Strichartz constant as in Lemma 2.2.
In particular, if k(u0,u1)kHsc×Hsc−1 6 δ, then the solution u is global and scatters.
Proof. We apply the Banach fixed point argument to prove this lemma. First we define
the solution map
t
(2.12) Φ(u(t)) = K˙(t)u +K(t)u − K(t−s)f(u(s))ds
0 1
Z0
on the complete metric space B
B = u∈ C(I;Hsc) : (u,u ) 6 2CE,kuk 6 2δ
t L∞(I;Hsc×Hsc−1) [W](I)
t
with the met(cid:8)ric d(u,v) = u−(cid:13)v (cid:13) . (cid:9)
(cid:13) [W]((cid:13)I)∩L∞t Hxsc
It suffices to prove that the operator defined by the RHS of (2.12) is a contraction
(cid:13) (cid:13)
map on B for I. If u∈ B,(cid:13)then b(cid:13)y Strichartz estimate (2.7), (2.8) and (2.10), we have
Φ(u) 6 K˙(t)u +K(t)u +C u3
[W](I) 0 1 [W](I) [W]∗(I)
(cid:13) (cid:13) (cid:13) 1+ 4 (cid:13)2(d−3) (cid:13) (cid:13)
(cid:13) (cid:13) 6δ(cid:13)+C u d−1 u(cid:13) d−1 . (cid:13) (cid:13)
[W](I) L∞H˙sc
t
Plugging the assumption kukL∞(I;H(cid:13)(cid:13)sc)(cid:13)(cid:13)6 2C(cid:13)(cid:13)E(cid:13)(cid:13)and kuk[W](I) 6 2δ, we see that for
u ∈B,
Φ(u) 6δ+C(2δ)1+d−41(2CE)2(dd−−13).
[W](I)
Thus we can choose δ(cid:13)small(cid:13)depending on E and the Strichartz constant C such that
(cid:13) (cid:13)
Φ(u) 6 2δ.
[W](I)
Similarly, if u∈ B, then (Φ(u),∂(cid:13)Φ(u))(cid:13) 6 2CE. Hence Φ(u) ∈ B for
(cid:13)t (cid:13)L∞(I;Hsc×Hsc−1)
t
u ∈B. That is, the functional Φ maps the set B back to itself.
(cid:13) (cid:13)
On the other hand, b(cid:13)y a same argum(cid:13)ent as before and Lemma 2.4, we have for
u,v ∈ B,
d(Φ(u),Φ(v)) .Cku3−v3k[W]∗(I)
2 2(d−2)
616Cku−vk[W](I)∩L∞t Hsck(u,v)k[dW−1](I) (u,v) [Wd−](1I)∩L∞t Hsc
2 2(d−2) (cid:13) (cid:13)
616C(4δ)d−1(4CE+2δ) d−1 d(u,v) (cid:13) (cid:13)
8 CHANGXINGMIAOANDJIQIANGZHENG
which allows us to derive
1
d(Φ(u),Φ(v)) 6 d(u,v),
2
by taking δ small such that
2 2(d−2) 1
16C(4δ)d−1(4CE +2δ) d−1 6 .
2
A standard fixed point argument gives a unique solution u of (1.1) on I ×Rd which
satisfies the bound (2.11). (cid:3)
Using Theorem 2.1 as well as its proof, one easily derives the following local theory
for (1.1). We omit the standard detail here.
Theorem 2.2. Assume that d > 5, s = d − 1. Then, given (u ,u ) ∈ Hsc(Rd) ×
c 2 0 1
Hsc−1(Rd) and t ∈ R, there exists a unique maximal-lifespan solution u: I ×Rd → R
0
to (1.1) with initial data u(t ),u (t ) = u ,u . This solution also has the following
0 t 0 0 1
properties:
(cid:0) (cid:1) (cid:0) (cid:1)
(1) (Local existence) I is an open neighborhood of t .
0
(2) (Blowup criterion) If sup(I) is finite, then u blows up forward in time (in the
sense of Definition 1.2). If inf(I) is finite, then u blows up backward in time.
(3) (Scattering) If sup(I) = +∞ and u does not blow up forward in time, then
u scatters forward in time in the sense (1.6). Conversely, given (v ,v˙ ) ∈
+ +
Hsc(Rd)×Hsc−1(Rd) there is a unique solution to (1.1) in a neighborhood of
infinity so that (1.6) holds.
2.2. Perturbation lemma. In this part, we give the perturbation theory of the solu-
tion of (1.1) with the global space-time estimate.
With any real-valued function u(t,x), we associate the complex-valued function
~u(t,x) by
(2.13) ~u = h∇isc−1 h∇iu−iu˙ , u= ℜh∇i−sc~u,
where ℜz denotes the real part of(cid:0)z ∈ C. Th(cid:1)en the free and nonlinear Klein-Gordon
equations are given by
(2+1)u = 0 ⇐⇒ (i∂ +h∇i)~u = 0,
t
(2.14)
((2+1)u = −f(u)⇐⇒ (i∂t +h∇i)~u = −h∇isc−1f(h∇i−scℜ~u),
Lemma 2.5. Let I be a time interval, t ∈ I and ~u,w~ ∈ C(I;L2(Rd)) satisfy
0
(i∂ +h∇i)~u =−h∇isc−1 f(u)+eq(u)
t
(i∂t +h∇i)w~ =−h∇isc−1(cid:2)f(w)+eq(w)(cid:3)
for some function eq(u),eq(w). Assume that for(cid:2)some constant(cid:3)s M,E > 0, we have
(2.15) w 6 M,
ST(I)
(2.16) ~u + w~ (cid:13) (cid:13) 6 E,
L∞t L2x(I×Rd) L∞t(cid:13)L2x(cid:13)(I×Rd)
Let t ∈ I, and let (u(t )(cid:13),u(cid:13)(t )) be close t(cid:13)o ((cid:13)w(t ),w (t )) in the sense that
0 0(cid:13) (cid:13)t 0 (cid:13) (cid:13) 0 t 0
(2.17) γ 6 ǫ,
0 ST(I)
(cid:13) (cid:13)
(cid:13) (cid:13)
SCATTERING THEORY FOR ENERGY-SUPERCRITICAL KLEIN-GORDON EQUATION 9
where ~γ = eih∇i(t−t0)(~u − w~)(t ) and 0 < ǫ < ǫ = ǫ (M,E) is a small constant.
0 0 1 1
Assume also that we have smallness conditions
(2.18) (eq(u),eq(w)) 6 ǫ,
ST∗(I)
where ǫ is as above and (cid:13) (cid:13)
(cid:13) (cid:13)
ST∗(I) = [W]∗(I)⊕L1(I;Bsc−1(Rd)).
t 2,2
Then we conclude that
u−w 6C(M,E)ǫ,
ST(I)
(2.19)
(cid:13) u(cid:13) 6C(M,E).
(cid:13) (cid:13)ST(I)
Proof. Since kwk 6M, there e(cid:13)xi(cid:13)sts a partition of the right half of I at t :
ST(I) (cid:13) (cid:13) 0
t <t < ··· < t , I = (t ,t ), I ∩(t ,∞) = (t ,t ),
0 1 N j j j+1 0 0 N
such that N 6 C(L,δ) and for any j =0,1,··· ,N −1, we have
(2.20) kwk 6δ ≪ 1.
ST(Ij)
The estimate on the left half of I at t is analogue, we omit it.
0
Let
(2.21) γ(t) = u(t)−w(t), ~γ (t) = eih∇i(t−tj)~γ(t ), 0 6 j 6 N −1,
j j
then ~γ satisfies the following difference equation
(i∂ +h∇i)~γ = −h∇isc−1 γ(γ2 +3γω+3ω2) +eq(u)−eq(w)
t
(~γ(tj) =~γj(tj), (cid:16)(cid:2) (cid:3) (cid:17)
which implies that
t
~γ(t) =~γ (t)+i eih∇i(t−s) h∇isc−1 γ(γ2+3γω+3ω2) +eq(u)−eq(w) ds,
j
Ztj n (cid:16) (cid:17)o
tj+1 (cid:2) (cid:3)
~γ (t) =~γ (t)+i eih∇i(t−s) h∇isc−1 γ(γ2+3γω+3ω2) +eq(u)−eq(w) ds.
j+1 j
Ztj n (cid:16) (cid:17)o
(cid:2) (cid:3)
By Strichartz estimate (2.7) and nonlinear estimate (2.8), we have
(2.22) kγ −γjkST(Ij)+kγj+1−γjkST(R)
. γ3+3γ2ω+3γω2 + (eq(u),eq(w))
[W]∗(Ij) ST∗(Ij)
(cid:13) 1+ 4 2(d−3) (cid:13) (cid:13) 4 2((cid:13)d−3)
.k(cid:13)γk d−1kγk d−1 (cid:13)+kωk (cid:13)kγkd−1 kγk (cid:13)d−1
ST(Ij) L∞t H˙sc ST(Ij) ST(Ij) L∞t H˙sc
4 2(d−3)
+kγk kωkd−1 kωk d−1 + (eq(u),eq(w)) .
ST(Ij) ST(Ij) L∞t H˙sc ST∗(Ij)
Therefore, assuming that (cid:13) (cid:13)
(cid:13) (cid:13)
(2.23) kγk 6 δ ≪ 1, ∀ j = 0,1,··· ,N −1,
ST(Ij)
then by (2.20) and (2.22), we have
(2.24) kγk +kγ k 6 Ckγ k +ǫ,
ST(Ij) j+1 ST(tj+1,tN) j ST(tj,tN)
10 CHANGXINGMIAOANDJIQIANGZHENG
for some absolute constant C > 0. By (2.17) and iteration on j, we obtain
δ
(2.25) kγk 6 (2C)Nǫ 6 ,
ST(I)
2
provided we choose ǫ sufficiently small. Hence the assumption (2.23) is justified by
1
continuity in t and induction on j. Then repeating the estimate (2.22) once again, we
can get the ST-norm estimate on γ, which implies the Strichartz estimates on u. (cid:3)
3. Profile decomposition
In this section, we first recall the linear profile decomposition of the sequence of
L2-bounded solutions of (i∂ +h∇i)~v = 0 which was established in [14]. And then we
x t
show the nonlinear profile decomposition which will be used to construct the critical
element and obtain its compactness properties in the next section.
3.1. Linear profile decomposition. First, we give some notation. For any triple
(tj,xj,hj) ∈ R × Rd × (0,1] with arbitrary suffix n and j, let τj, Tj, and h∇ij
n n n n n n
respectively denote the scaled time shift, the unitary and the self-adjoint operators in
L2(Rd), defined by
j j
t x−x
(3.1) τnj = − nj , Tnjϕ(x) = (hjn)−d2ϕ j n , h∇ijn = −∆+(hjn)2.
h h
n (cid:16) n (cid:17) q
Now we can state the linear profile decomposition as follows
Lemma 3.1 (Linear profile decomposition, [14]). Let ~v (t) = eih∇it~v (0) be a sequence
n n
of free Klein-Gordon solutions with uniformly bounded L2(Rd)-norm. Then after re-
x
placing it with some subsequence, there exist K ∈ {0,1,2...,∞} and, for each integer
j ∈ [0,K), ϕj ∈ L2(Rd) and {(tjn,xjn,hjn)}n∈N ⊂ R×Rd×(0,1] satisfying the following.
Define ~vj and ω~k for each j < k 6 K by
n n
k−1
(3.2) ~v (t,x) = ~vj(t,x)+~ωk(t,x),
n n n
j=0
X
where
(3.3) ~vnj(t,x) = eih∇i(t−tjn)Tnjϕj(x) = Tnj eih∇ijnt−hjntjnϕj ,
(cid:16) (cid:17)
then we have
(3.4) lim lim ~ωk = 0,
k→Kn→∞ n L∞t (R;B∞−,d2∞(Rd))
and for any l < j < k 6 K and any(cid:13)(cid:13)t ∈(cid:13)(cid:13)R,
(3.5) lim µ~vl,µ~vj 2 = 0 = lim µ~vj,µ~ωk 2 ,
n→∞ n n L2x n→∞ n n L2x
hl (cid:10) hj |(cid:11)tj −tk|+|xj −(cid:10)xk| (cid:11)
(3.6) lim n + n + n n n n = +∞,
n→∞ hj hl hl
(cid:26)(cid:12) n(cid:12) (cid:12) n(cid:12) n (cid:27)
where µ ∈ MC and MC(cid:12)is d(cid:12)efin(cid:12)ed t(cid:12)o be
(cid:12) (cid:12) (cid:12) (cid:12)
MC = µ = F−1µ˜F| µ˜ ∈ C(Rd),∃ lim µ˜(x) ∈ R .
|x|→∞
n o