Global existence for a coupled wave system related to the Strauss conjecture

0.43 MB·

by  Jason Metcalfe

GLOBAL EXISTENCE FOR A COUPLED WAVE SYSTEM
RELATED TO THE STRAUSS CONJECTURE
JASONMETCALFEANDDAVIDSPENCER
7
1
0 Abstract. Acoupledsystemofsemilinearwaveequationsisconsidered,and
2 asmalldataglobalexistenceresultrelatedtotheStraussconjectureisproved.
Previousresultshaveshownthatoneofthepowersmaybereducedbelowthe
n
criticalpowerfortheStraussconjectureprovidedtheotherpowersufficiently
a
exceeds such. The stability of such results under asymptotically flat pertur-
J
bations of the space-time where an integrated local energy decay estimate is
9 availableisestablished.
1

]
P
A 1. Introduction
.
h The purpose of this article is to establish global existence for a coupled system
t of wave equations, which is related to the Strauss conjecture, on asymptotically
a
m flat space-times that permit a localized energy estimate. It is now well-known that
nonlinear wave equations
[

(cid:3)u:=(∂2−∆)u=F (u), (t,x)∈R+×Rn
1 t p
v where
2
(cid:88)
4 (1.1) |u|j|∂jF (u)|(cid:46)|u|p for u small
u p
6
0≤j≤2
5
0 haveglobalsolutionsforsufficientlysmallinitialdataprovidedp>pc wherepc >1
. solves
1
0 (1.2) (n−1)p2−(n+1)p −2=0.
c c
7
1 Moreover, blow up is known to occur for p < p . These results originated in [23]
√ c
: for n = 3 where p = 1 + 2, and following [44] the problem became known
v c
i as the Strauss conjecture. Global existence in general dimension was eventually
X
established in [20], [45]; see the references therein for many intermediate results.
r Blowupbelowthecriticalexponentwasprovedin[40]. Seealso[39],[51]forfurther
a
results at the critical exponent.
In the current work, we shall examine a system of the form
(cid:3)u=|v|p, (cid:3)v =|u|q.
In the flat case, the coupled system was examined in [16], and it was shown that
globalexistencemaybeestablishedforpowersinthenonlinearitybelowthecritical
Date:January23,2017.
The first author was supported in part by NSF grant DMS-1054289. The second author
was supported in part by a Summer Undergraduate Research Fellowship (SURF) through the
University of North Carolina, and the results contained herein were developed as a part of his
UndergraduateHonorsThesis.
1
2 JASONMETCALFEANDDAVIDSPENCER
exponent provided the power on the coupled equation exceeds the same. Indeed,
setting
(cid:110)q+2+p−1 p+2+q−1(cid:111) n−1
(1.3) C(p,q)=max , − ,
pq−1 pq−1 2
it was shown that small data global existence holds for C(p,q) < 0 and that such
fails for C(p,q) > 0. Notice that C(p,p) = 0 corresponds precisely to (1.2). In
particular, note that small data global existence may be established for powers
p<p provided that the other power q sufficiently exceeds p . In addition to [16],
c c
see [17], [14] for treatments of C(p,q)>0 and [1], [15], [25], and [21] for analysis of
the critical curve C(p,q) = 0. Moreover, see the overview [26] of this and related
problems.
Here we seek to establish the same using techniques that are sufficiently robust
so as to allow background geometries. Specifically, we shall use a variant of the
weighted Strichartz estimates of [22], [19], which were further developed in [27],
[35], and the localized energy estimate to prove such global existence.
We shall examine operators of the form
(1.4) Pu=∂ gαβ∂ u+bα∂ u+cu
α β α
onspace-timesM whereM =R ×R3 orM =R ×(R3\K)whereKhasasmooth
+ +
boundary and K ⊂ {x : |x| < R }. Here g is a Lorentzian metric, and we make
0
the assumption that g can be written as
(1.5) g(t,x)=m+g (t,r)+g (t,x)
0 1
where m = diag(−1,1,1,1) is the Minkowski metric. The components g and
0
g will represent long-range and short-range perturbations respectively. They are
1
asymptotically flat in the sense that
(1.6) (cid:107)∂µ g (cid:107) =O(1), i=0,1, |µ|≤3.1
t,x i,αβ (cid:96)i+|µ|L∞
1 t,x
Due to the need to commute with spatial rotations, the long-range perturbation
g is assumed to be spherically symmetric in the sense that the coefficients only
0
(spatially) depend on r =|x| and
(1.7)
g−g =(−1+g˜ (t,r))dt2+2g˜ (t,r)dtdr+(1+g˜ (t,r))dr2+(1+g˜ (t,r))r2dω2 .
1 00 01 11 22 S2
and, by (1.6), (cid:107)∂µ g˜ (cid:107) = O(1) for |µ| ≤ 3. The coefficients of the lower-
t,x αβ (cid:96)|µ|L∞
1 t,x
order perturbations decay are assumed to decay as follows:
(1.8) (cid:107)∂µ b(cid:107) +(cid:107)∂µ c(cid:107) =O(1), |µ|≤2.
t,x (cid:96)1+|µ|L∞ t,x (cid:96)2+|µ|L∞
1 t,x 1 t,x
We shall also assume that the perturbations admit a (weak) localized energy
decay. More specifically, we assume that there is R (with R > R in the case
1 1 0
1Here,asin[35],foranormA,weset
(cid:107)u(cid:107)(cid:96)sqA=(cid:13)(cid:13)(cid:13)2js(cid:107)φj(x)u(t,x)(cid:107)A(cid:13)(cid:13)(cid:13)(cid:96)q , (cid:88)φ2j(x)=1, suppφj ⊂{(cid:104)x(cid:105)≈2j}.
j≥0 j≥0
COUPLE WAVE EQUATIONS ON ASYMPTOTICALLY FLAT BACKGROUNDS 3
that M =R ×(R3\K)) so that if u solves Pu=F then
+
(1.9) (cid:107)∂∂µu(cid:107) +(cid:107)(1−χ)∂∂µu(cid:107) +(cid:107)∂µu(cid:107)
L∞t L2x (cid:96)−∞1/2L2t,x (cid:96)−∞3/2L2t,x
(cid:88)
(cid:46)(cid:107)u(0, ·)(cid:107) +(cid:107)∂ u(0, ·)(cid:107) + (cid:107)∂νF(cid:107)
H|µ|+1 t H|µ| L1L2
t x
|ν|≤|µ|
for all |µ|≤2. Here χ is a smooth function that is identically 1 on B :={|x|≤
R1/2
R /2} and is supported on B .
1 R1
On (1+3)-dimensional Minkowski space (i.e. when g ≡ g ≡ 0), it is known
0 1
that
(cid:107)∂u(cid:107) +(cid:107)∂u(cid:107) +(cid:107)u(cid:107) (cid:46)(cid:107)∂u(0, ·)(cid:107) +(cid:107)(cid:3)u(cid:107) ,
L∞t L2x (cid:96)∞−1/2L2t,x (cid:96)∞−3/2L2t,x L2 L1tL2x
whichisastrongerversionoftheµ=0estimateabove. Andastheflatd’Alembertian
commutes with ∂ , the higher order variants readily follow. Such estimates orig-
t,x
inated in [36]. They follow, e.g., by multiplying (cid:3)u by a multiplier of the form
r ∂ u+ n−1 1 u, integrating over [0,T]×R3, and integrating by parts. See,
r+2j r 2 r+2j
e.g., [43], [29]. And see, e.g., [34] for a more complete history. These estimates
are known to be rather robust in the asymptotically flat regime. Even without the
cutoff, they are known to hold for small, possibly time-dependent perturbations of
Minkowski space [29, 30], [33, 32], [2] and for time-independent nontrapping per-
turbationsintheproductmanifoldsettingdueto,e.g.,[8],[6],[42]. See[31]forthe
most general results in the nontrapping regime.
Thepresenceoftrappedraysisaknownobstructiontothelocalizedenergyesti-
mate [37], [38]. The asymptotic flatness restricts the possibility of trapped rays to
a compact set, and when the trapping is sufficiently weak, a localized energy esti-
mate where one, say, cuts off away from the trapping may sometimes be recovered.
Allowing for this is the reason for the cutoff in assumption (1.9). Previous results
havethenverified(1.9)inanumberofsettingswheretrappingoccurs,includingon
the Schwarzschild space-time [4, 5], [10, 11], [28], on Kerr space-times with a(cid:28)M
[47] (see also [3], [9, 12] for some closely related results and [13] for a related result
that holds for the full subextremal range |a|<M), and on certain warped product
manifolds that contain degenerate trapping [7].
We now introduce the specific problem at hand. With two possibly different
operators P and P subject to hypotheses (1.5)-(1.9), we examine the coupled
1 2
system
P u=F (v), P v =F (u),
1 p 2 q
(1.10) u(0,x)=f (x), v(0,x)=f (x),
1 2
∂ u(0,x)=g (x), ∂ v(0,x)=g (x).
t 1 t 2
Here F and F are two functions satisfying (1.1).
p q
Forthesystem(1.10),weshallestablishthefollowingsmalldataglobalexistence
result:
Theorem1.1. SupposethatP andP areoperatorsoftheform (1.4)sothat (1.5),
1 2
(1.6), (1.7), (1.8), and (1.9) hold. Moreover assume that 2<p,q and C(p,q)<0.
4 JASONMETCALFEANDDAVIDSPENCER
Figure 1. Range of allowable indices
q
4.0
3.5
3.0
2.5
p
2.0 2.5 3.0 3.5 4.0
Then if f ,g ,f ,g ∈C∞2 and
1 1 2 2 c
(1.11) (cid:107)(f ,f )(cid:107) +(cid:107)(g ,g )(cid:107) ≤ε
1 2 H3 1 2 H2
with ε sufficiently small, there exists a global solution (u,v) to (1.10).
We note that the techniques to prove Theorem 1.1 also work in four spatial
dimensions, but as they require p,q ≥ 2 and p = 2 in this case, nothing new is
c
gained over [35].
This result follows a number of studies that established various existence results
related to the Strauss conjecture in the presence of background geometry. In ex-
terior domains, these included [18], [22], [41], and [52]. And on asymptotically flat
backgrounds, see[42], [49], [27], [35], and[48]. See, also, theexpositoryarticle[50].
A key component of many of these results is the weighted Strichartz estimate of
[22],[19]. Here,inparticular,werelyonthevariantofthatdevelopedin[35],which
is based on the local energy estimates of [33].
Figure1demonstratestherangeofallowableindices. Theorem1.1allowsforany
pair of indices (p,q) that land outside of the shaded region in the lower left corner.
The techniques of, e.g., [35], however, only trivially apply in the rectangular region
√
where p,q >p =1+ 2. The other curve will be discussed further in Section 3.
c
The method that we shall employ is similar in spirit to that of [35], which in
turn is based on a number of preceding works. Near infinity, where the asymptotic
flatness allows us to think of the geometry as a small perturbation of Minkowski
space, variants of the weighted Strichartz estimates of [22], [19] are employed. The
assumed localized energy estimate handles the remaining compact region, where
the geometry has the most significant role. It also allows the analyses done in the
two regions to be glued together. Such a strategy has become common. See, e.g.,
[27], [28], [33, 32], [34], [46].
2Forsimplicityofexposition,wehavetakenthedataheretobecompactlysupported,butthis
maybereplacedbyaconditionsuchas[35,(5.3)].
COUPLE WAVE EQUATIONS ON ASYMPTOTICALLY FLAT BACKGROUNDS 5
2. Main Estimates
Beforeweproceedtothemainestimatesforthelinearequation,wefirstintroduce
some notation. We shall be using a restricted set of the classical invariant vector
fields. We let Ω = x ∂ −x ∂ denote the generators of spatial rotations. And
ij i j j i
we let Y = {∇ ,Ω} and Z = {∂,Ω}, where ∂ = (∂ ,∇ ) denotes the space-time
x t x
gradient. We also introduce the shorthand |Z≤mu| = (cid:80) |Zµu| for summing
|µ|≤m
over multi-indices of order ≤ m. A similar notation where the absolute values are
replaced by a norm shall also be used. Finally, for the mixed norms that appear in
the weighted Strichartz estimates below, we fix the convention
(cid:104)(cid:90) (cid:16)(cid:90) (cid:104)(cid:90) (cid:105)q/s (cid:17)p/q (cid:105)1/p
(cid:107)f(cid:107)LptLqrLsω = |f(t,rω)|sdωS2 r2dr dt ,
with the obvious changes for Lebesgue indices of ∞.
The main linear estimate that will be applied near infinity, where the operators
may be viewed as small perturbations of the flat d’Alembertian, is the following
weighted Strichartz estimate:
Theorem 2.1 ([35]). Suppose that P is anoperator of theform (1.4) satisfying the
hypotheses (1.5), (1.6), (1.7), (1.8), and (1.9). Suppose that w(0, ·) and ∂ w(0, ·)
t
are compactly supported and satisfy (1.11). Then there exists R > R so that for
2 1
any R > R if ψ is identically 1 on {|x| ≥ 2R} and vanishes on {|x| < R}, then
2 R
we have
(2.1) (cid:107)ψ Z≤2w(cid:107) ≤C ε+C (cid:107)ψ Y≤1Pw(0, ·)(cid:107)
R (cid:96)p23−p4−sLptLprL2ω 1 1 R H˙s−1
+C (cid:107)ψp˜Z≤2Pw(cid:107) +C (cid:107)∂≤2Pw(cid:107)
1 R (cid:96)−12−sL1L1L2 1 L1tL2rL2ω
1 t r ω
and
(2.2) (cid:107)ψ w(cid:107) (cid:46)ε+(cid:107)ψp˜Pw(cid:107) +(cid:107)Pw(cid:107)
R (cid:96)p23−p4−sLptLprL2ω R (cid:96)−1 12−sL1tL1rL2ω L1tL2rL2ω
for any p∈(2,∞), s∈(1/2−1/p,1/2), and p˜>0.
These estimates originated in [22] and [19] for the flat d’Alembertian. The
above version is essentially from [35], which in particular allows for asymptotically
flatoperatorsanddoesnotnecessitatecompactlysupportedinitialdatathoughwe
assume that here for simplicity. These estimates follow by interpolating a variant
of the localized energy estimate with a trace theorem on the sphere. We note a
few minor modifications from the version in [35]: (1) We have allowed a different
power of the cutoff function in the right sides; (2) We have more carefully stated
the requirements on R so that in the sequel we are able to use the same R for both
u and v; (3) We have stated separately the case where no vector fields are applied
since this is used when showing that our iteration is Cauchy. These all follow from
trivial modifications of the proof of [35].
A key to tying the region near infinity to the remaining compact region where
(1.9) is the primary tool is the following weighted Sobolev inequalities. These are
variants of the original estimates of [24] and follow by localizing, applying Sobolev
embeddings on R×S2, and adjusting the volume elements to match those of R3 in
polar coordinates. See, e.g., [27], [35] for proofs.
6 JASONMETCALFEANDDAVIDSPENCER
Lemma 2.2. On R3, for R≥1, β ∈R, and 2≤p≤q ≤∞, we have
(2.3) (cid:107)rβu(cid:107)LqrL∞ω(r≥R+1) (cid:46) (cid:88) (cid:107)rβ−p2+q2Yµu(cid:107)LprL2ω(r≥R).
|µ|≤2
If 2≤p≤q ≤4 and β ∈R, we also have
(2.4) (cid:107)rβu(cid:107)LqrL4ω(r≥R+1) (cid:46) (cid:88) (cid:107)rβ−p2+q2Yµu(cid:107)LprL2ω(r≥R).
|µ|≤1
3. Small data global existence
(cid:16) (cid:17)
WenowproveTheorem1.1. Weshallapply(2.1)touwith(p,s)= q,7+4p−3pq
2−2pq
(cid:16) (cid:17)
and to v with (p,s) = p,7+4q−3pq . The requirement from Theorem 2.1 that
2−2pq
s> 1 − 1 then corresponds to
2 p
2+p+q−1 2+q+p−1
<1, <1
pq−1 pq−1
respectively. In the case of n=3, this produces the requirement that C(p,q)<0.
We shall assume, without loss of generality, that (p,q) satisfy
(3.1) p(q−2)<3, q(p−2)<3,
which corresponds to the condition that s < 1/2 in Theorem 2.1. It is these
conditions that are represented by the curve in Figure 1, below which they are
satisfied. Intheunshadedregiontotheleftoftherectangle,if(p,q)satisfyC(p,q)<
0 but one of the above conditions is violated, one may simply choose any q˜<q so
that C(p,q˜) < 0 and so that both conditions in (3.1) hold. One simply imagines
|u|q = |u|q−q˜|u|q˜ and argues as below with the exponent q˜ replacing q. Simple
Sobolev embeddings control the remaining q−q˜powers. In the unshaded region
belowtherectangle,onearguessimilarlybyreducingthepowerofp. Intheshaded
rectangle, the methods of [35] apply directly and no further argument is needed.
Let s = 7+4p−3pq, s = 7+4q−3pq, α = 3 − 4 −s , and α = 3 − 4 −s . We
1 2−2pq 2 2−2pq 1 2 q 1 2 2 p 2
note that the power of the weight in the right side of (2.1) then satisfies:
1 1
− −s =pα , − −s =qα .
2 1 2 2 2 1
We solve (1.10) via an iteration, and at this point, the arguments that are used
areakintothoseof[22],[27],and[35]. Settingu ≡0andv ≡0,werecursively
−1 −1
define u ,v , j ≥0 to solve
j j
P u =F (v ), P v =F (u ),
1 j p j−1 2 j q j−1
(3.2) u (0,x)=f (x), v (0,x)=f (x),
j 1 j 2
∂ u (0,x)=g (x), ∂ v (0,x)=g (x).
t j 1 t j 2
We introduce the quantity
(3.3) Mk(u,v)=(cid:107)ψRZ≤ku(cid:107)(cid:96)αq1LqtLqrL2ω +(cid:107)ψRZ≤kv(cid:107)(cid:96)αp2LptLprL2ω
+(cid:107)∂≤k(u,v)(cid:107) +(cid:107)∂≤k∂(u,v)(cid:107) .
(cid:96)−∞3/2L2tL2rL2ω L∞t L2rL2ω
COUPLE WAVE EQUATIONS ON ASYMPTOTICALLY FLAT BACKGROUNDS 7
Our first goal is to inductively show that M (u ,v ) ≤ 4C ε for some uniform
2 j j 2
constant C . We first note that from (1.11) (and the assumption that the data are
2
compactly supported) we easily obtain
(cid:107)ψ Y≤1P u (0, ·)(cid:107) +(cid:107)ψ Y≤1P v (0, ·)(cid:107) ≤Cε.
R 1 j H˙s1−1 R 2 j H˙s2−1
And thus, for C chosen large enough, by (2.1),
2
(3.4) (cid:107)ψRZ≤2uj(cid:107)(cid:96)αq1LqtLqrL2ω +(cid:107)ψRZ≤2vj(cid:107)(cid:96)αp2LptLprL2ω ≤C2ε
+C(cid:107)ψRpZ≤2Fp(vj−1)(cid:107)(cid:96)pα2L1L1L2 +C(cid:107)ψRqZ≤2Fq(uj−1)(cid:107)(cid:96)qα1L1L1L2
1 t r ω 1 t r ω
+C(cid:107)∂≤2F (v )(cid:107) +C(cid:107)∂≤2F (u )(cid:107) .
p j−1 L1L2L2 q j−1 L1L2L2
t r ω t r ω
And from (1.11) and (1.9), we have
(3.5) (cid:107)∂≤2(u ,v )(cid:107) +(cid:107)∂≤2∂(u ,v )(cid:107) ≤C ε
j j (cid:96)−3/2L2L2L2 j j L∞L2L2 2
t r ω t r ω
+C(cid:107)∂≤2F (v )(cid:107) +C(cid:107)∂≤2F (u )(cid:107) .
p j−1 L1L2L2 q j−1 L1L2L2
t r ω t r ω
From these, we first notice that M (u ,v ) ≤ 2C ε, which provides the base case
2 0 0 2
for the induction.
We then assume that M (u ,v )≤4C ε and show that M (u ,v )≤4C ε.
2 j−1 j−1 2 2 j j 2
We first notice that (1.1) gives
(3.6) |Z≤2F (u)|(cid:46)|u|p−1|Z≤2u|+|u|p−2|Z≤1u|2.
p
BytheSobolevembeddingsH2 ⊂L∞ andH1 ⊂L4 onS2 andH¨older’sinequality,
ω ω ω ω
this gives
(cid:107)Z≤2F (v )(cid:107) (cid:46)(cid:107)v (cid:107)p−1(cid:107)Z≤2v (cid:107) +(cid:107)v (cid:107)p−2(cid:107)Z≤1v (cid:107)2
p j−1 L2ω j−1 L∞ω j−1 L2ω j−1 L∞ω j−1 L4ω
(cid:46)(cid:107)Z≤2v (cid:107)p .
j−1 L2
ω
It then immediately follows that
(3.7)
(cid:107)ψRpZ≤2Fp(vj−1)(cid:107)(cid:96)p1α2L1tL1rL2ω (cid:46)(cid:107)ψRZ≤2vj−1(cid:107)p(cid:96)αp2LptLprL2ω (cid:46)(M2(uj−1,vj−1))p (cid:46)εp.
Thelastinequalityresultsfromtheinductivehypothesis. Asimilarargumentshows
that
(3.8) (cid:107)ψRqZ≤2Fq(uj−1)(cid:107)(cid:96)qα1L1L1L2 (cid:46)εq.
1 t r ω
To finish the proof of the boundedness of M (u ,v ), it remains to examine the
2 j j
L1L2L2 pieces in the right sides of (3.4) and (3.5). The analyses will be done
t r ω
separately outside of a ball of radius 2R +1, where the resulting terms will be
compared to the weighted Strichartz portions of M (u ,v ), and inside the
2 j−1 j−1
remaining compact set where the localized energy portions will be used.
WebeginwiththeformerusingtheweightedSobolevinequalitiesofLemma2.2.
Starting again at (3.6), we have
(3.9)
(cid:107)Z≤2Fp(vj−1)(cid:107)L1tL2r≥2R+1L2ω (cid:46)(cid:107)r−pα−21vj−1(cid:107)pL−pL12pp(p−−21)L∞(cid:107)rα2ψRZ≤2vj−1(cid:107)LptLprL2ω
t r≥2R+1 ω
(cid:13) (cid:0) (cid:1) (cid:13)p−2
+(cid:13)(cid:13)rp−22 −α2−p2+12 vj−1(cid:13)(cid:13)LpL∞ L∞(cid:107)rα2+p2−12Z≤1vj−1(cid:107)2LptL4r≥2R+1L4ω.
t r≥2R+1 ω
8 JASONMETCALFEANDDAVIDSPENCER
Since (3.1) guarantees that s <1/2, it follows that pα =−1 −s ≥−1, which is
1 2 2 1
equivalent to
α 2 p−2
− 2 − + ≤α .
p−1 p p(p−1) 2
Thus, (2.3) yields
(cid:107)r−pα−21vj−1(cid:107) 2p(p−1) (cid:46)(cid:107)rα2Z≤2vj−1(cid:107)LpLp L2.
LpL p−2 L∞ t r≥2R ω
t r≥2R+1 ω
The estimate (2.4) directly applies to yield
(cid:107)rα2+p2−12Z≤1vj−1(cid:107)LpL4 L4 (cid:46)(cid:107)rα2Z≤2vj−1(cid:107)LpLp L2.
t r≥2R+1 ω t r≥2R ω
As above, pα ≥−1, which implies that
2
2 (cid:16) 2 1(cid:17) 2
−α − + − ≤α .
p−2 2 p 2 p 2
Hence (2.3) gives
(cid:13) (cid:0) (cid:1) (cid:13)
(cid:13)(cid:13)rp−22 −α2−p2+21 vj−1(cid:13)(cid:13) (cid:46)(cid:107)rα2Z≤2vj−1(cid:107)LpLp L2.
LpL∞ L∞ t r≥2R ω
t r≥2R+1 ω
Plugging each of these bounds into (3.9) and using that ψ is identically 1 on
R
r ≥2R, it follows that
(3.10)
(cid:107)Z≤2F (v )(cid:107) (cid:46)(cid:107)ψ Z≤2v (cid:107)p (cid:46)(M (u ,v ))p (cid:46)εp.
p j−1 L1tL2r≥2R+1L2ω R j−1 (cid:96)αp2LptLprL2ω 2 j−1 j−1
And an analogous argument in the symmetric variable q shows that
(3.11) (cid:107)Z≤2F (u )(cid:107) (cid:46)εq,
q j−1 L1L2 L2
t r≥2R+1 ω
which leaves the analysis of the L1L2L2 pieces over the region r ≤ 2R+1 where
t r ω
the coefficients of Z are bounded.
We again start with (3.6) and apply H¨older’s inequality to obtain
(3.12) (cid:107)Z≤2F (v )(cid:107)
p j−1 L1L2 L2
t r≤2R+1 ω
(cid:46)(cid:107)v (cid:107)p−2 (cid:107)v (cid:107) (cid:107)∂≤2v (cid:107)
j−1 L∞t L∞r L∞ω j−1 L2tL∞r≤2R+1L∞ω j−1 L2tL2r≤2R+1L2ω
+(cid:107)v (cid:107)p−2 (cid:107)∂≤1v (cid:107)2 .
j−1 L∞L∞L∞ j−1 L2L4 L4
t r ω t r≤2R+1 ω
By Sobolev embeddings, we have
(cid:107)v (cid:107) (cid:46)(cid:107)∂≤1v (cid:107) (cid:46)(cid:107)∂≤1∂v (cid:107) .
j−1 L∞L∞L∞ j−1 L∞L6L6 j−1 L∞L2L2
t r ω t r ω t r ω
Similarly, Sobolev embeddings (with a localizing factor) give
(cid:107)v (cid:107) (cid:46)(cid:107)∂≤2v (cid:107) (cid:46)(cid:107)∂≤2v (cid:107)
j−1 L2tL∞r≤2R+1L∞ω j−1 L2tL2r≤2R+2L2ω j−1 (cid:96)∞−3/2L2tL2rL2ω
and
(cid:107)∂≤1v (cid:107) (cid:46)(cid:107)∂≤2v (cid:107) (cid:46)(cid:107)∂≤2v (cid:107) .
j−1 L2tL4r≤2R+1L4ω j−1 L2tL2r≤2R+2L2ω j−1 (cid:96)−∞3/2L2tL2rL2ω
These bounds in (3.12) show that
(cid:107)Z≤2F (v )(cid:107) (cid:46)(cid:107)∂≤1∂v (cid:107)p−2 (cid:107)∂≤2v (cid:107)2
(3.13) p j−1 L1tL2r≤2R+1L2ω j−1 L∞t L2rL2ω j−1 (cid:96)−∞3/2L2tL2rL2ω
(cid:46)(M (u ,v ))p (cid:46)εp.
2 j−1 j−1
COUPLE WAVE EQUATIONS ON ASYMPTOTICALLY FLAT BACKGROUNDS 9
And an analogous argument gives
(3.14) (cid:107)Z≤2F (u )(cid:107) (cid:46)εq.
q j−1 L1L2 L2
t r≤2R+1 ω
Using (3.7), (3.8), (3.10), (3.11), (3.13), and (3.14) in (3.4) and (3.5), we obtain
M (u ,v )≤2C ε+C εmin(p,q)
2 j j 2 3
for some constant C that is independent of j. Since p,q ≥ 2, if ε is sufficiently
3
small, the desired bound M (u ,v )≤4C ε follows.
2 j j 2
It remains to show that the sequence converges. To do so, we shall show
1
(3.15) M (u −u ,v −v )≤ M (u −u ,v −v ),
0 j j−1 j j−1 2 0 j−1 j−2 j−1 j−2
and this will complete the proof. Applying (2.2) and (1.9), we have
(3.16) (cid:107)ψR(uj −uj−1)(cid:107)(cid:96)αq1LqtLqrL2ω +(cid:107)ψR(vj −vj−1)(cid:107)(cid:96)αp2LptLprL2ω
(cid:46)(cid:107)ψRp(Fp(vj−1)−Fp(vj−2)(cid:107)(cid:96)pα2L1L1L2 +(cid:107)ψRq(Fq(uj−1)−Fq(uj−2))(cid:107)(cid:96)qα1L1L1L2
1 t r ω 1 t r ω
+(cid:107)F (v )−F (v )(cid:107) +(cid:107)F (u )−F (u )(cid:107) ,
p j−1 p j−2 L1L2L2 q j−1 q j−2 L1L2L2
t r ω t r ω
and
(3.17) (cid:107)(u −u ,v −v )(cid:107) +(cid:107)∂(u −u ,v −v )(cid:107)
j j−1 j j−1 (cid:96)−3/2L2L2L2 j j−1 j j−1 L∞L2L2
t r ω t r ω
(cid:46)(cid:107)F (v )−F (v )(cid:107) +(cid:107)F (u )−F (u )(cid:107) .
p j−1 p j−2 L1L2L2 q j−1 q j−2 L1L2L2
t r ω t r ω
Noting that
(3.18) |F (v )−F (v )|(cid:46)(|v |p−1+|v |p−1)|v −v |,
p j−1 p j−2 j−1 j−2 j−1 j−2
we can quickly observe that
(3.19)
(cid:107)ψRp(Fp(vj−1)−Fp(vj−2)(cid:107)(cid:96)pα2L1L1L2
1 t r ω
(cid:16) (cid:17)
(cid:46) (cid:107)ψRvj−1(cid:107)p(cid:96)αp−21LptLprL∞ω +(cid:107)ψRvj−2(cid:107)p(cid:96)αp−21LptLprL∞ω (cid:107)ψR(vj−1−vj−2)(cid:107)(cid:96)αp2LptLprL2ω
(cid:16) (cid:17)
(cid:46) (cid:107)ψ Z≤2v (cid:107)p−1 +(cid:107)ψ Z≤2v (cid:107)p−1
R j−1 (cid:96)αp2LptLprL2ω R j−2 (cid:96)αp2LptLprL2ω
×(cid:107)ψR(vj−1−vj−2)(cid:107)(cid:96)αp2LptLprL2ω
(cid:46)(4C ε)p−1M (u −u ,v −v ).
2 0 j−1 j−2 j−1 j−2
A similar argument yields
(3.20)
(cid:107)ψRq(Fq(uj−1)−Fq(uj−2)(cid:107)(cid:96)qα1L1L1L2 (cid:46)(4C2ε)q−1M0(uj−1−uj−2,vj−1−vj−2).
1 t r ω
We also start at (3.18) to control the L1L2L2 terms. There we get, using (2.3)
t r ω
as above,
(3.21)
(cid:107)F (v )−F (v )(cid:107)
p j−1 p j−2 L1L2 L2
t r≥2R+1 ω
(cid:46)((cid:107)r−pα−21vj−1(cid:107)p−12p(p−1) +(cid:107)r−pα−21vj−2(cid:107)p−12p(p−1) )(cid:107)rα2ψR(vj−1−vj−2)(cid:107)LptLprL2ω
LpL p−2 L∞ LpL p−2 L∞
t r≥2R+1 ω t r≥2R+1 ω
(cid:46)[(M (u ,v ))p−1+(M (u ,v ))p−1]M (u −u ,v −v )
2 j−1 j−1 2 j−2 j−2 0 j−1 j−2 j−1 j−2
(cid:46)εp−1M (u −u ,v −v ).
0 j−1 j−2 j−1 j−2
10 JASONMETCALFEANDDAVIDSPENCER
And similarly
(3.22) (cid:107)F (u )−F (u )(cid:107) (cid:46)εq−1M (u −u ,v −v ).
q j−1 q j−2 L1L2 L2 0 j−1 j−2 j−1 j−2
t r≥2R+1 ω
Finally, using Sobolev embeddings as in (3.13),
(3.23)
(cid:107)F (v )−F (v )(cid:107)
p j−1 p j−2 L1L2 L2
t r≤2R+1 ω
(cid:46)[(cid:107)v (cid:107)p−2 (cid:107)v (cid:107) +(cid:107)v (cid:107)p−2 (cid:107)v (cid:107) ]
j−1 L∞t L∞r L∞ω j−1 L2tL∞r≤2R+1L∞ω j−2 L∞t L∞r L∞ω j−2 L2tL∞r≤2R+1L∞ω
×(cid:107)v −v (cid:107)
j−1 j−2 L2L2 L2
t r≤2R+1 ω
(cid:46)[(M (u ,v ))p−1+(M (u ,v ))p−1]M (u −u ,v −v )
2 j−1 j−1 2 j−2 j−2 0 j−1 j−2 j−1 j−2
(cid:46)εp−1M (u −u ,v −v ),
0 j−1 j−2 j−1 j−2
and by the same procedures
(3.24) (cid:107)F (u )−F (u )(cid:107) (cid:46)εq−1M (u −u ,v −v ).
q j−1 q j−2 L1L2 L2 0 j−1 j−2 j−1 j−2
t r≤2R+1 ω
Plugging (3.19)-(3.24) into (3.16) and (3.17) immediately yields (3.15) provided
that ε is sufficiently small, and this completes the proof.
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