Scattering above energy norm of solutions of a focusing logloglog energy-supercritical Schrodinger equation with radial data below ground state

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by  Tristan Roy

SCATTERING ABOVE ENERGY NORM OF SOLUTIONS OF A
FOCUSING LOGLOGLOG ENERGY-SUPERCRITICAL
SCHRO¨DINGER EQUATION WITH RADIAL DATA BELOW
7
1 GROUND STATE
0
2
TRISTANROY
n
a
J
Abstract. Given n ∈ {3,4} and k > n2, we prove scattering of the ra-
0 dial H˜k := H˙k(Rn)∩H˙1(Rn)− solutions of the logloglog focusing energy-
1 supercriticalSchro¨dingerequationi∂tu+△u=−|u|n−42ulogγlog(log(1010+|u|2))
for a range of positive γs, for energies below that of the ground states, and
]
P for potentials below that of the ground states. The proof uses in particular
A arguments of[2,9,11,13,14].
.
h
t
a
m
1. Introduction
[

We shall study the radial solutions of the following Schr¨odinger equation in
1
dimension n, n∈{3,4}:
v
9
8
6 (1) i∂tu+△u =−|u|n−42ug(|u|)
2
with g(|u|):=logγlog(log(1010+|u|2)) and γ >0.
0
. This equation has many connections with the following power-type Schr¨odinger
1
equation, p>1
0
7
1
(2) i∂ v+△v =−|v|p−1v
: t
v
i (2) has a natural scaling: if v is a solution of (2) with data v(0):=v and if λ∈R
X 0
isaparameterthenv (t,x):= 1 u t ,x isalsoasolutionof(2)butwithdata
λ 2 λ2 λ
r λp−1
a vλ(0,x) := 12 u0 λx . If sp := n2 −(cid:0)p−21 th(cid:1)en the H˙sp norm of the initial data is
λp−1
invariantunderthe(cid:0)sca(cid:1)ling: thisiswhy(2)issaidtobeH˙sp-critical. Ifp=1+ 4
n−2
then (2) is H˙1 (or energy) critical. The energy-critical Schr¨odinger equation
(3) i∂tu+△u =−|u|n−42u
has received a great deal of attention. Cazenave and Weissler [3] proved the local
well-posedness of (3): given any u(0) such that ku(0)k < ∞ there exists, for
H˙1
2(n+2) 2(n+2)
some t close to zero, a unique u∈C([0,t ],H˙1)∩L n−2 L n−2 ([0,t ]) satisfying
0 0 t x 0
(3) in the sense of distributions
(4) u(t) =eit△u(0)+i tei(t−t′)△ |u(t′)|n−42u(t′) dt′·
0
R 1 h i
2 TRISTANROY
The next step is to understand the asymptotic behavior of the solutions of (3).
It is well-known that (3) has a family of stationary solutions (the ground states)
Wλ,θ(x):=eiθ n1−2W λx thatsatisfy△Wλ,θ+|Wλ,θ|n−42Wλ,θ =0withθ ∈[0,2π]
λ 2
and W defined by
(cid:0) (cid:1)
W(x) := 1 ·
n−2
1+ |x|2 2
(cid:16) n(n−2)(cid:17)
The asymptotic behavior of the solutions for energies below that of the ground
states has been studied in [7]. In particular global existence and scattering (i.e
the linear asymptotic behavior) were proved for potential energies below that of
the ground states. The asymptotic behavior of the solutions was studied in [5] for
energies equal to that of the ground states and in [9] for energies slightly larger
than that of the ground states.
If p > 1 + 4 then s > 1 and we are in the energy supercritical regime.
n−2 p
Since for all ǫ > 0 there exists cǫ > 0 such that |u|n−42u . |u|n−42ug(|u|) ≤
cǫmax(1,||u|n−42+ǫu|)thenthenonlinearityof(1)is(cid:12)(cid:12)saidtob(cid:12)(cid:12)eba(cid:12)(cid:12)relysupercriti(cid:12)(cid:12)cal.
In this paper we study the asymptotic behavior(cid:12)of H˜k-so(cid:12)lutio(cid:12)ns of (1) for n(cid:12) ∈
{3,4}. Recall the local-wellposed result:
Proposition 1. “Local well-posedness ” [11] Let n∈{3,4} and let k > n. Let
2
M be such that ku k ≤ M. Then there exists ǫ := ǫ(M) > 0 small such that if
0 H˜k
T >0 (T := time of local existence) satisfies
l l
keit△u k ≤ǫ
(5) 0 2(n+2) 2(n+2)
Ltn−2 Lxn−2 ([0,Tl])
then there exists a unique
2(n+2) 2(n+2) 2(n+2) 2(n+2)
(6) u∈C([0,Tl],H˜k)∩Ltn−2 Lxn−2 ([0,Tl])∩D−1Lt n Lx n ([0,Tl])
2(n+2) 2(n+2)
∩Lt n D−kLx n ([0,Tl])
such that
(7) u(t)=eit△u0+i 0tei(t−t′)△ |u(t′)|n−42u(t′)g(|u(t′)|) dt′
is satisfied in the sense of disRtributions.(cid:16)Here D−αLr := H˙α,r e(cid:17)ndowed with the
norm kfk :=kDαfk .
D−αLr Lr
This allows to define the notion of maximal time interval of existence I :=
max
(T ,T ), that is the union of all the intervals I containing 0 such that (7) holds in
− +
2(n+2) 2(n+2) 2(n+2) 2(n+2) 2(n+2) 2(n+2)
theclassC(I,H˜k)∩Ltn−2 Lxn−2 (I)∩D−1Lt n Lx n (I)∩D−kLt n Lx n (I).
Recall the following proposition:
Proposition 2. “Global well-posedness: criterion” [11] If |I |<∞ then
max
kuk =∞
(8) 2(n+2) 2(n+2)
Ltn−2 Lxn−2 (Imax)
FOCUSING LOGLOGLOG SUPERCRITICAL RADIAL SCHRO¨DINGER EQUATION 3
Remark 1. Proposition 1 and 2 were proved in [11] for solutions of barely super-
critical defocusing nonlinearities. The proof can be easily adapted to solutions of
(1).
With this in mind, global well-posedness follows from an a priori bound of the
form kuk ≤ f(T,ku k ) for arbitrarily large time T > 0. In
2(n+2) 2(n+2) 0 H˜k
Ltn−2 Lxn−2 ([−T,T])
factforsomedataweshallprovethatthebounddoesnotdependontimeT,which
will show scattering.
Before stating the main theorem, we recallsome generalnotation. We write a≪b
if the value of a is much smaller that that of b, a ≫ b if the value of a is much
larger than that of b, and a∼b if a≪b and b≪a are not true. We say that C˜ is
the constant determined by a . b if it the smallest constant C such that a ≤ Cb.
We write a = o(b) if there exists a constant 0 < c ≪ 1 such that |a| ≤ c|b|. We
define b+ = b+ǫ for 0 < ǫ ≪ 1. If b+ appears in a mathematical expression
such as a ≤ Cb+, then we ignore the dependance of C on ǫ in order to make our
presentation simple.
Let 2∗ := 2n . If f ∈H˜k then we define the energy
n−2
(9) E(f) := 1 |∇f(x)|2− F(f,f¯)(x)dx,
2 Rn Rn
with R R
(10) F(z,z¯) := |z|snn−+22g(s)ds·
0
Indeed R
(11) F(f,f¯)(x)dx .kfk2∗ g(kfk ).kfk2∗ g(kfk ):
Rn L2∗ L∞ H˙1 H˜k
this follows(cid:12)fRrom a simple int(cid:12)egration by part
(cid:12) (cid:12)
(12) F(z,z¯) ∼|z|n2−n2g(|z|),
combined with the Sobolev inequality
(13) kukL∞t L∞x (J) .kukL∞t H˜k(J)·
If f ∈H˙1 then we denote by E˜(f) the following
1 1
(14) E˜(f):= |∇f(x)|2− |f(x)|2∗ dx·
2ZRn 2∗ ZRn
Hence E˜(f)=E(f)+X(f) with
X(f):= |f(x)|(g(s)−1)s2∗−1 dsdx·
Rn 0
We define the functional
R R
K˜(f):=k∇fk2 −kfk2∗ ·
L2 L2∗
Let u be an H˜k solution of (1). A simple computation shows that the energy
E(u(t))is conserved,or,inotherwords,thatE(u(t))=E(u ). Letχbe a smooth,
0
4 TRISTANROY
radial function supported on |x| ≤ 2 such that χ(x) = 1 if |x| ≤ 1. If x ∈ Rn,
0
R>0, then we define the mass within the ball B(x ,R)
0
1
(15) Mass(B(x ,R),u(t)) := |u(t,x)|2dx 2
0 B(x0,R)
(cid:16) (cid:17)
Recall (see [13]) that R
(16) Mass(B(x ,R),u(t)) .R sup k∇u(t′)k
0 t′∈[0,t] L2
and that its derivative satisfies
′
(17) |∂ Mass(u(t),B(x ,R))| . supt′∈[0,t]k∇u(t )kL2
t 0 R
The main result of this paper is:
Theorem 3. Let δ > 0, n ∈ {3,4}, and u ∈ H˜k := H˙k(Rn)∩H˙1(Rn),k > n,
0 2
radial such that 1
(18) ku0kH˜k &1, E(u0)<(1−2δ)E˜(W), andku0kL2∗ <kWkL2∗·
Let a := 365 (resp. a := 683 ) if n =3 (resp. n =4). There exist C ≫1 such
n n 6 a
that if γ >0 satisfies
1
(19) logγloglogCaCaCaδ2(αα−1)+ ku0k2H˜k−1≪δ
 
with α := 1 then the solution of (1) with data u(0) := u exists for all time T.
γan 0
Moreover there exists a scattering state u ∈H˜k such that
0,+
(20) tl→im∞ku(t)−eit△u0,+kH˜k =0
and there exists C :=C(ku k ,δ) such that kuk ≤C.
0 H˜k 2(n+2) 2(n+2)
Ltn−2 Lxn−2 (R)
By symmetry an analogous statement holds for negative times.
Remark 2. Notice that ku0kL2∗ = kWkL2∗ is impossible. Indeed, recall [1, 10]
the sharp Sobolev inequality kfkL2∗ ≤ C∗k∇fkL2 with C∗ := kk∇WWkLk2L∗2. Hence
F(y)<(1−δ)F(kWkL2∗) with y :=ku0kL2∗ and
1 1
(21) F(y):= C2y2− y2∗·
2 ∗ 2∗
Remark 3. Observe from (13) and (19) that kg(u )−1k ≪δ. Hence E˜(u )<
0 L∞ 0
(1−δ)E˜(W). Hence 0<δE˜(W)< 1 k∇Wk2 −k∇u k2 . So in particular δ is
2 L2 0 L2
bounded from above by a constant that does not depend on ku k .
(cid:0) (cid:1) 0 H˜k
Remark 4. For data satisfying (18), Theorem 3 implies global regularity since by
the Sobolev embedding kukL∞t L∞x (R) .kukL∞t H˜k(R) for k > n2.
1If we only assume that ku0kH˜k ≪1 then the same conclusion holds. This is a consequence
ofthelocaltheory: seeAppendix.
FOCUSING LOGLOGLOG SUPERCRITICAL RADIAL SCHRO¨DINGER EQUATION 5
We recall some standard inequalities. The following Sobolev inequality holds:
kuk .kDuk ·
(22) 2(n+2) 2(n+2) 2(n+2) 2n(n+2)
Ltn−2 Lxn−2 (J) Ltn−2 Lxn2+4 (J)
Ifuis asolutionofi∂ u+△u=G, u(t=0):=u onJ suchthatu(t)∈H˜k, t∈J,
t 0
then the Strichartz estimates (see for example [6]) yield
kuk +kDjuk +kDjuk
L∞H˙j(J) 2(n+2) 2(n+2) 2(n+2) 2n(n+2)
(23) t Lt n Lx n (J) Ltn−2 Lxn2+4 (J)
.kDjGk +ku k
2(n+2) 2(n+2) 0 H˙j
Ltn+4 Lxn+4 (J)
if j ∈{1,k}. We define
(24)
Q(J,u):=kuk +kDuk +kDkuk +kuk ·
L∞H˜k(J) 2(n+2) 2(n+2) 2(n+2) 2(n+2) 2(n+2) 2(n+2)
t Lt n Lx n (J) Lt n Lx n (J) Ltn−2 Lxn−2 (J)
We also recall the following proposition:
Proposition 4. “A fractional Leibnitz rule”[11] Let 0 ≤ α ≤ 1, k and β
integers such that k ≥ 2 and β ≥ 1, (r,r ,r ) ∈ (1,∞)3, r ∈ (1,∞] be such that
1 2 3
1 = β + 1 + 1 . Let F :R+ →R be a Ck- function and let G:=C×C→C be a
r r1 r2 r3
Ck- function such that
F[i](x) =O F(x) , τ ∈[0,1]: |F[j](τx+(1−τ)y)|.|F[j](x)|+|F[j](y)|
xi
and (cid:16) (cid:17)
O(|x|β+1−i), i≤β+1
(25) |G[i](x,x¯)|=
0, i>β+1
(cid:26)
for 0≤i≤k and 1≤j ≤k. Then
(26) Dk−1+α(G(f,f¯)F(|f|) .kfkβ kDk−1+αfk kF(|f|)k
Lr Lr1 Lr2 Lr3
Here F[i] a(cid:13)nd G[i] denote the ith- d(cid:13)erivatives of F and G respectively.
(cid:13) (cid:13)
Now we explain how this paper is organized. In Section 2 we prove the main
result of this paper, i.e Theorem 3. The proof relies upon the following bound of
kuk on an arbitrarily long time interval
2(n+2) 2(n+2)
Ltn−2 Lxn−2
2(n+2) 2(n+2)
Proposition 5. “ Bound of L2(n−2)L2(n−2) norm ” Let u be an H˜k solution of
t x
(1) such that (18) holds. Let J :=[t ,t ] be an interval. Assume that for all t∈J
1 2
(27) g(u(t))−1≪δ·
ThenthereexistsaconstantC ≫1suchthatifkuk ≤M forsomeM ≫1,
0 L∞H˜k(J)
t
then
2(n+2) Cδ−21gan+(M)
(28) kuk n−2 ≤C 0
2(n+2) 2(n+2) 0
Ltn−2 Lxn−2 (J)
6 TRISTANROY
ThisboundprovedonanarbitrarytimeintervalJ,combinedwithalocalinduc-
tionontime ofsome Strichartzestimates, allowsto controla posteriori the L∞H˜k
t
norm of the solution and some other norms at H˜k regularity on J, and to show a
posteriori thatthecondition(27)holdsonJ,assumingthatg growsslowlyenough,
in the sense of (19): see [11, 12] for a similar argument. Global well-posedness and
scattering of H˜k-solutions of (1) follow easily from the finiteness of these bounds.
In Section 3, we prove Proposition 5. We mention the main differences between
this paper and [11]. First one has to assume the condition (27). This condition,
combined with the energy conservation law and the variational properties of the
groundstates,assurethat somerelevantnorms(suchas the kinetic energyandthe
potentialenergy)areboundedonJ,sothatwecanapplythetechniquesofconcen-
tration (see e.g [2, 13]) in order to prove (28). Roughly speaking, we divide J into
2(n+2) 2(n+2)
subintervals(J ) suchthatthe L n−2 L n−2 normofuconcentrates,i.e itis
l 1≤l≤L t x
smallbutalsosubstantial. Ourgoalistoestimatethenumberofthesesubintervals.
It is already known that the mass on a ball centered at the origin concentrates for
alltime ofeachof these subintervals. In [11], a Morawetz-typeestimate (combined
withthemassconcentration)wasusedtoprovethatthefollowingstatementholds:
one of these subintervals is large compare with J. In this paper we use the virial
identity and we adapt arguments in [9, 14] to prove a decay at some time of the
potential energy on a ball centered at the origin, which leads to a contradiction
unless the statement above holds. With the use of this statement one can show
that there exists a significantnumber of subintervals (in comparisonwith the total
numberofsubintervals)thatconcentratearoundsometimeandsuchthatthemass
concentrates around the origin, which yields an estimate of the number of all the
subintervals. The process involves several estimates. One has to understand how
they depend on g(M) and δ since this will play an important role in the choice of
γ (see(19))forwhichwe haveglobalwell-posednessandscatteringofH˜k-solutions
of (1).
2. Proof of Theorem 3
The proof is made of two steps:
• finite bound of kuk , kuk , kDuk
L∞H˜k(R) 2(n+2) 2(n+2) 2(n+2) 2(n+2)
t Ltn−2 Lxn−2 (R) Lt n Lx n (R)
and kDkuk . By time reversal symmetry 2 and by mono-
2(n+2) 2(n+2)
Lt n Lx n (R)
tone convergence it is enough to find, for all T ≥ 0, a finite bound of all
these norms restricted to [0,T] and the bound should not depend on T.
(29) F := T ∈[0,∞):sup Q([0,t],u)≤M ; g(u(t))−1≪δ
t∈[0,T] 0
n o
We claim that F = [0,∞) for M , a large constant (to be chosen later)
0
depending only on ku k and δ. Indeed
0 H˜k
– 0∈F.
– F is closed by continuity
2i.eift→u(t,x)isasolutionof(1)thent→u¯(−t,x)isalsoasolutionof(1)
FOCUSING LOGLOGLOG SUPERCRITICAL RADIAL SCHRO¨DINGER EQUATION 7
– F is open. Indeed let T ∈F. Then, by continuity there exists δ′ > 0
such that for T′ ∈ [0,T +δ′] we have Q([0,T′]) ≤ 2M . In view of
0
(28), this implies, in particular, that
2(n+2) Cδ−12gan+(2M0)
(30) kuk n−2 ≤C 0
2(n+2) 2(n+2) 0
Ltn−2 Lxn−2 ([0,T′])
LetJ :=[0,a]be aninterval. We getfrom(23), (13), andProposition
4
(31)
4
Q(J,u).ku k + kDuk +kDkuk kukn−2
0 H˜k 2(n+2) 2(n+2) 2(n+2) 2(n+2) 2(n+2) 2(n+2)
(cid:18) Lt n Lx n (J) Lt n Lx n (J)(cid:19) Ltn−2 Lxn−2 (J)
g kuk
L∞H˜k(J)
t
.(cid:16)ku k +Q((cid:17)J,u)kukn−42 g(Q(J,u))
0 H˜k 2(n+2) 2(n+2)
Ltn−2 Lxn−2 (J)
LetC be the constantdetermined by . inthe secondline of(31). We
may assume WLOG that C ≫1. Let 0<ǫ≪1. Notice that if J sat-
isfies kuk = ǫ then a simple continuity
2(n+2) 2(n+2) n−2
Ltn−2 Lxn−2 (J) g 4 (2Cku0kH˜k)
argument shows that
(32) Q(J,u) ≤2Cku k ·
0 H˜k
Wedivide[0,T′]intosubintervals(J ) suchthatkuk =
i 1≤i≤I 2(n+2) 2(n+2)
ǫ ,1≤i<Iandkuk ≤Ltn−2 Lxǫn−2 (Ji) .
n−2 2(n+2) 2(n+2) n−2
g 4 ((2C)iku0kH˜k) Ltn−2 Lxn−2 (JI) g 4 ((2C)Iku0kH˜k)
Wewillseeshortlythatsuchapartitionexists. Firstweprovethe fol-
lowing claim:
I−1
Claim: Let X := 1 .
i=1 gn+22((2C)iku0kH˜k)
There exists C¯ ≫P1 such that if¯i:= C¯ku k 3, then
0 H˜k
(cid:2) (cid:3)
(33) X & I(n−+12−)γ¯i + (n+2¯i)γ
log 2 (¯i) log 2 log(¯i)
Proof. Wegetgn+22 (2C)iku0kH˜k ≈log(n+22)γ(log(log((2C)2iku0k2H˜k))).
Hence choosing C¯ ≫1 we see that
(cid:0) (cid:1)
¯i I−1
X & 1 + 1
(n+2)γ (n+2)γ
i=1log 2 log(¯i) ¯i+1 log 2 log(i)
=XP +X · P
1 2
3Here[x]denotes theentirepartofx
8 TRISTANROY
Let F(I) := I−1 1 dx. Observe that X ≥ F(I) &
¯i (n+2)γ 2
log 2 log(x)
I−1−¯i ,Rwhere the last estimate can be deduced easily from
(n+2)γ
log 2 log(I)
integrating F(I) once by parts. Hence (33) holds.
(cid:3)
We have
(34) CC0δ−12ga+n(2M0) &X & I−1−¯i + ¯i
0 (n+2)γ (n+2)γ
log 2 log(I) log 2 log(¯i)
Moreover,by iterating the procedure in (31) and (32) we get
(35) Q([0,T′],u) ≤(2C)Iku k ·
0 H˜k
Inviewof(35),wemayassumeWLOGthatI ≫¯i. Henceweseefrom
(34) that
loglog(I) .δ−12gan+(2M0)·
Since γ ≪ 1 we see that we can choose M := M (ku k ,δ) such
0 0 0 H˜k
that
log M0
(36) δ−12gan+(2M0) ≪loglog (cid:18)lokgu(02kCH˜)k(cid:19)·
 
This shows that Q([0,T′],u) ≤ M . We now find values of M for
0 0
which (36) holds. Let M0′ := kuM0k0H˜k. Choose M0 ≫ ku0kH˜k. Hence,
with this choice of M , it is sufficient to find M′ such that
0 0
logα−loglog(M0′) ≫ 1
logloglog(M0′) δα2−
which is satisfied if
1
M′ ≥C¯C¯C¯δ2(αα−1)+
0
for some well-chosen constant C¯ ≫ 1. Hence we see from (19) that
g(u(t))−1≪δ for t∈[0,T′].
• Scattering: it is enough to prove that e−it△u(t) has a limit as t → ∞ in
H˜k. If t <t then by dualizing (23) with G=0 we get
1 2
(37)
ke−it1△u(t1)−e−it2△u(t2)kH˜k
4
.kukn−2 kDkuk +kDuk g(M ),
2(n+2) 2(n+2) 2(n+2) 2(n+2) 2(n+2) 2(n+2) 0
Ltn−2 Lxn−2 ([t1,t2])(cid:18) Lt n Lx n ([t1,t2]) Lt n Lx n ([t1,t2])(cid:19)
and we conclude that there exists A(ǫ) such that if t ≥ t ≥ A(ǫ) then
2 1
ke−it1△u(t1)−e−it2△u(t2)kH˜k ≤ǫ. Hence scattering.
FOCUSING LOGLOGLOG SUPERCRITICAL RADIAL SCHRO¨DINGER EQUATION 9
3. Proof of Proposition 5
In this section we prove Proposition 5. First we prove a preliminary lemma.
3.1. A lemma.
Lemma 6. There exists δ′ ≈δ12 such that for all t∈J
(38) E˜(u(t))<(1−δ)E˜(W)
and
|∇u(t)|2 dx≤(1−δ′) |∇W|2 dx
Rn Rn
(39)
R R
K˜(u(t))≥δ′ |∇u(t)|2 dx
Rn
R
Proof. Recall that E˜(W):= 1 − 1 kWk2∗ . Define
2 2∗ L2∗
(cid:0) (cid:1)
F :={T ∈J : (39)holds fort∈[t ,T]}
1
We claim that F = J. Clearly t ∈ F and F is closed by continuity of the flow.
1
It remains to prove that it is open. By continuity there exists β > 0 such that
(39) holds for T′ ∈ [t ,T +β] with δ′ substituted for 2δ′. Hence in view of the
1
assumptions, the Sobolev embedding, and the conservation of the energy
(40) E˜(u(t))=E(u)+X(u(t))<(1−2δ)E˜(W)+δ <(1−δ)E˜(W)
Let us define
(a): ku(t)kL2∗ <kWkL2∗
(b): k∇u(t)k <k∇Wk
L2 L2
With(40)inminditislefttothereadertocheckthatiftsatisfies(a)thentsatisfies
(b). Recall [7] that
k∇u(t)k2 <k∇Wk2 ⇒ (39)holds
L2 L2
Hence T′ ∈F.
(cid:3)
3.2. The proof. We provenow Proposition5 by using this lemma andconcentra-
tion techniques (see e.g [2, 13]).
We divide the interval J =[t ,t ] into subintervals (J :=[t¯,t¯ ]) such that
1 2 l l l+1 1≤l≤L
2(n+2)
(41) kuk n2−(n2+2) 2(n+2) =η1
Ltn−2 Lxn−2 (Jl)
2(n+2)
(42) kuk n2−(n2+2) 2(n+2) ≤η1
Ltn−2 Lxn−2 (JL)
with c ≪ 1 and η := c1 . In view of (28), we may replace WLOG the
1 1 2(n+2)
g 6−n (M)
“≤′′ sign with the “=′′ sign in (42).
10 TRISTANROY
Recallthe notion of exceptionalintervals and the notion of unexceptionalintervals
(such a notion appears in the study of (3) in [13]). Let
c η g−1(M) 22, n=3
(43) η := 2 1
2 1
( c2(cid:0)η135g−28(M(cid:1)) 3 , n=4
with c ≪c . An interval J =[t¯(cid:0),t¯ ] of the(cid:1)partition (J ) is exceptional
2 1 l0 l0 l0+1 l 1≤l≤L
if
2(n+2) 2(n+2)
(44) kul,t1k n2−(n2+2) 2(n+2) +kul,t2k n2−(n2+2) 2(n+2) ≥η2·
Ltn−2 Lxn−2 (Jl0) Ltn−2 Lxn−2 (Jl0)
In view of (22) and (23), we have
(45) card{J : J exceptional} .η−1·
l l 2
Recall that on each unexceptional subintervals J there is a ball for which we have
l
a mass concentration.
Result 1. “Mass concentration” [11] There exists an x ∈ Rn, two constants
l
0<c≪1 and C ≫1 such that for each unexceptional interval J and for t∈J
l l
• if n=3
(46) Mass u(t),B(xl,Cg133(M)|Jl|12) ≥cg−133(M)|Jl|21
(cid:16) (cid:17)
• if n=4
(47) Mass u(t),B(xl,Cg137(M)|Jl|12) ≥cg−137(M)|Jl|21
(cid:16) (cid:17)
The radialsymmetry allowsto provethat, in fact, there is a mass concentration
around the origin on each unexceptional interval J . Recall
l
Result2. “Mass concentration around the origin”[11]Thereexistapositive
constant ≪1 (that we still denote by c to avoid too much notation) and a constant
C˜ ≫1 such that on each unexceptional interval J we have
l
• if n=3
(48) Mass u(t),B(0,C˜g1639(M)|Jl|12) ≥cg−133(M)|Jl|21
(cid:16) (cid:17)
• if n=4
(49) Mass u(t),B(0,C˜g51(M)|Jl|21) ≥cg−137(M)|Jl|21
(cid:16) (cid:17)
Remark 5. Notice that the values of the parameters η and η are not chosen
1 2
randomly. They are the largest ones such that all the constraints appearing in the
proof of Result 1 and Result 2 are satisfied.