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Scattering problem for the generalized Korteweg-de Vries equation

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Abstract

In this paper we study the scattering problem for the initial value problem of the generalized Korteweg-de Vries (gKdV) equation. The purpose of this paper is to achieve two primary goals. Firstly, we show small data scattering for (gKdV) in the weighted Sobolev space, ensuring the initial and the asymptotic states belong to the same class. Secondly, we introduce two equivalent characterizations of scattering in the weighted Sobolev space. In particular, this involves the so-called conditional scattering in the weighted Sobolev space. A key ingredient is incorporation of the scattering criterion for (gKdV) in the Fourier-Lebesgue space by the authors into the the scattering problem in the weighted Sobolev space.
arXiv:2408.00261v1 [math.AP] 1 Aug 2024
SCATTERING PROBLEM FOR THE GENERALIZED
KORTEWEG-DE VRIES EQUATION
SATOSHI MASAKI AND JUN-ICHI SEGATA
Abstract. In this paper we study the scattering problem for the initial
value problem of the generalized Korteweg-de Vries (gKdV) equation.
The purpose of this paper is to achieve two primary goals. Firstly, we
show small data scattering for (gKdV) in the weighted Sobolev space,
ensuring the initial and the asymptotic states belong to the same class.
Secondly, we introduce two equivalent characterizations of scattering in
the weighted Sobolev space. In particular, this involves the so-called
conditional scattering in the weighted Sobolev space. A key ingredient
is incorporation of the scattering criterion for (gKdV) in the Fourier-
Lebesgue space by the authors [30] into the the scattering problem in
the weighted Sobolev space.
1. Introduction
In this paper we study the scattering problem for the generalized Korteweg-
de Vries (gKdV) equation
(1.1) tu+3
xu=µ∂x(|u|2αu), t, x R
under the initial condition
(1.2) u(0, x) = u0(x), x R,
where u:R×RRis an unknown function, u0:RRis a given
function, and µR\{0}and α > 0 are constants. We call that (1.1) is
defocusing if µ > 0 and focusing if µ < 0. Equation (1.1) is a generalization
of the Korteweg-de Vries equation which models long waves propagating in
a channel [26] and the modified Korteweg-de Vries equation which describes
a time evolution for the curvature of certain types of helical space curves
[27].
Equation (1.1) has the following conservation laws: If u(t) is a solution
to (1.1) on the time interval Iwith 0 I, then, u(t) has conservation of the
mass
M[u(t)] := 1
2ku(t, ·)k2
L2=M[u0](1.3)
and conservation of the energy
E[u(t)] := 1
2kxu(t, ·)k2
L2+µ
2α+ 2 ku(t, ·)k2α+2
L2α+2 =E[u0](1.4)
for any tI.
2000 Mathematics Subject Classification. Primary 35Q53, 35B40; Secondary 35B30.
Key words and phrases. generalized Korteweg-de Vries equation, scattering problem.
1
2 SATOSHI MASAKI AND JUN-ICHI SEGATA
We take the initial data u0from the weighted Sobolev space H1H0,1,
where H1is the usual Sobolev space and H0,1is the weighted L2space
defined by
H0,1=H0,1(R) := {fL2(R) ; kfkH0,1=khxifkL2<∞}
with hxi=p1 + |x|2. By the Sobolev embedding, one sees that the weighted
space H1H0,1is embedded into Lrˆ
Lrfor any r[1,], where ˆ
Lris
the Fourier-Lebesgue space defined for 1 6r6by
ˆ
Lr=ˆ
Lr(R) := {f S(R) ; kfkˆ
Lr=kˆ
fkLr<∞}(1.5)
and rdenotes the older conjugate of r.
The purpose of this paper is to achieve two primary goals. Firstly, we
show small data scattering for (1.1)-(1.2) in the weighted Sobolev space,
ensuring the initial and the asymptotic states belong to the same class.
Secondly, we introduce two equivalent characterizations of scattering in the
weighted Sobolev space. In particular, this involves the so-called conditional
scattering in the weighted Sobolev space.
There are many results on the small data scattering problem for (1.1).
Strauss [35] proved that if α > (3+ 21)/4, and u0L(2α+2)/(2α+1) ,xu0
L2are sufficiently small, then the solution to (1.1) is global and scatters in
H1. Ponce and Vega [34] have shown a similar scattering result for α > (5 +
73)/8. Christ and Weintein [3] improved their results to α > (1957)/8.
Furthermore, Hayashi and Naumkin extended their results to α>1, where
they proved an usual scattering for (1.1) when α > 1 [12] (see also ote
[5] for construction of large data wave operator) and a modified scattering
for α= 1 [13, 14, 15] (See also Harrop-Griffiths [11], Germain, Pusateri
and Rousset [9], Correia, ote, and Vega [4] for other approaches). In
those results, the classes of the initial states and the asymptotic states are
different.
Form the physical perspective, it is natural that the initial and the as-
ymptotic states belong to the same class. For this direction, Kenig, Ponce
and Vega [20] proved the small scattering of (1.1) in the scaling critical
space ˙
Hsαfor α>2, where sα:= 1/21 is a scaling critical exponent
(see also Strunk [36]). Since the scaling critical exponent sαis negative in
the mass-subcritical case α < 2, the scattering of (1.1) in the scaling crit-
ical space ˙
Hsαbecomes rather a difficult problem. Tao [37] proved global
well-posedness and scattering for small data for (1.1) with the quartic non-
linearity µ∂x(u4) in ˙
Hs3/2. Later on, Koch and Marzuola [25] simplified
Tao’s proof and extended his result to a Besov space ˙
Bs3/2
,2. In [30], the au-
thors proved small data scattering for (1.1) in the framework of the scaling
critical Fourier-Lebesgue space ˆ
Lαfor 8/5< α 62.
For the large initial data, Dodson [7] has shown the global well-posedness
and scattering in L2for (1.1) with the defocusing and mass-critical non-
linearity (i.e., µ > 0 and α= 2) by using the concentration compactness
argument by Kenig and Merle [18] and the monotonicity formula for (1.1)
by Tao [38] (see also Killip, Kwon, Shao and Vi¸san [21] for the existence of
the minimal non-scattering solution for (1.1) with the focusing, mass-critical
nonlinearity). After that Farah, Linares, Pastor and Visciglia [8] proved the
SCATTERING FOR THE GENERALIZED KDV EQUATION 3
global well-posedness and scattering in H1for (1.1) with the defocusing and
mass-supercritical nonlinearity (i.e., µ > 0 and α > 2) by adapting the
the concentration compactness argument into H1. For the mass-subcritical
case α < 2, the authors [31, 32] proved the existence of the minimal non-
scattering solution for (1.1) with 5/3< α < 2 by applying the concentration
compactness argument in the Fourier-Bourgain-Morrey space. Furthermore,
Kim [24] proved the conditional scattering in the Fourier-Bourgain-Morrey
space for (1.1) when the nonlinear term is defocusing and mass-subcritical
with 5/3< α < 2. Note that for the case α= 1, it is well-known that (1.1) is
completely integrable. By using the inverse scattering method, Deift-Zhou
[6] obtained asymptotic behavior in time of solution to (1.1) with α= 1 and
without smallness on the initial data.
1.1. Local well-posedness in a weighted space. In this paper, we use
several notions of a solution to (1.1). Let {V(t)}tRbe a unitary group
generated by the 3
x. For an interval IR, we define
S(I) := {u:I×RR;kukS(I)<∞},(1.6)
kukS(I):= kukL
5
2α
x(R;L5α
t(I)).
Definition 1.1 (a solution to (1.1)).Let X=ˆ
Lα,X=H1ˆ
Lα, or
X=H1H0,1. For an interval IR, we say a function u:I×RR
is a X-solution on Iif V(t)u(t)C(I;X),kukS(J)<for any compact
JI, and the identity
(1.7) V(t2)u(t2) = V(t1)u(t1) + Zt2
t1
V(τ)x(|u|2αu)(τ)
holds for any t1, t2I.
Due to the modification in the definition of a solution, a natural extension
of the initial condition (1.2) to an arbitrary time t0Ris as follows:
(1.8) V(t0)u(t0) = V(t0)u0X.
Remark 1.2.V(t) is an isometry on ˆ
Lαor H1ˆ
Lα. Hence, V(t)u(t)
C(I;X) is equivalent to u(t)C(I;X) when X=ˆ
Lαor X=H1ˆ
Lα.
Moreover, (1.7) is equivalent to the validity of the standard Duhamel for-
mula. Furthermore, (1.8) is equivalent to u(t0) = u0X. However, V(t)
is not a bounded operator from H1H0,1to itself for any t6= 0 and hence
these modifications are essential in the case X=H1H0,1. We also remark
that the embedding
H1H0,1֒H1ˆ
Lα֒ˆ
Lα
holds for any 1 6α6. Hence, a H1H0,1-solution is a H1ˆ
Lα-solution
and similarly a H1ˆ
Lα-solution is a ˆ
Lα-solution. Further, it is known that
ˆ
Lα-solution is unique if 8/5< α < 10/3 (see [30, Theorem 1.2]).
Before the scattering problem, let us consider the local well-posedness. It
is noteworthy that the local well-posedness in ˆ
Lαand H1ˆ
Lαare already
established in [30]. We also have the local well-posedness in the weighted
Sobolev space H1H0,1.
4 SATOSHI MASAKI AND JUN-ICHI SEGATA
Theorem 1.3 (Local well-posedness in H1H0,1).The initial value problem
(1.1) under (1.8) is locally well-posed in the weighted Sobolev space H1H0,1.
More precisely, suppose that V(t0)u0H1H0,1for some t0R. Then,
there exist a interval It0and a unique H1H0,1-solution u(t)to (1.1)
under (1.8) on Isuch that
kV(t)ukL
t(I;H1
xH0,1
x).kV(t0)u0kH1
xH0,1
x+ht0iku0k2α+1
H1
x.
Moreover, the data-to-solution map V(t0)u07→ uis a continuous map from
H1H0,1to L(I;H1H0,1).
Now, we turn to the global existence of a solution. To this end, we
introduce the notion of the maximal lifespan of a solution. For a X-solution
u(t) to (1.1) on an interval I, we define
Tmax := sup{TR;u:X-solution to (1.1) on [t0, T ]},
Tmin := inf{TR;u:X-solution to (1.1) on [T, t0]}
with a picked t0I. Note that these quantities are independent of the
choice of t0I. Further, we refer Imax = (Tmin, Tmax ) to as the maximal
lifespan of a solution u. A solution uon Imax is referred to as a maximal-
lifespan solution. We say a solution uis global for positive time direction
(resp. negative time direction) if Tmax =(resp. Tmin =−∞).
It is obvious from the definition that Imax depends on the choice of the
notion of a solution, i.e., on X. However, those with X=ˆ
Lαand X=
H1ˆ
Lαcoincides each other. This property, which is called the persistence
of H1-regularity, implies that if a ˆ
Lα-solution usatisfies u(t)H1at some
time in its maximal lifespan (as a ˆ
Lα-solution) then u(t)H1holds in
the whole maximal lifespan and further uis a H1ˆ
Lα-solution with the
same maximal lifespan. Our next result shows that Imax is also the same for
H1H0,1-solution.
Theorem 1.4 (Blowup alternative).Let ube a maximal-lifespan H1H0,1-
solution and Imax = (Tmin, Tmax )be its maximal lifespan as a H1H0,1-
solution. If Tmax <then
lim
TTmax0kukS([t0,T )) =.
A similar alternative holds for Tmin. In particular, Imax is the same as those
as a ˆ
Lα- and H1ˆ
Lα-solution.
This property reads as the persistence of the boundedness V(t)u(t)
H1H0,1for ˆ
Lα-solutions. Due to this property, we use the notation Imax
without clarifying the notion of a solution.
1.2. Main results. Now, we consider the scattering problem. We give the
definition of scattering in X.
Definition 1.5. Let X=ˆ
Lα,X=H1ˆ
Lα, or X=H1H0,1. We say a
X-solution u(t)scatters in Xfor positive time direction if Tmax = +and
there exists a unique function u+Xsuch that
lim
t+kV(t)u(t)u+kX= 0.(1.9)
The scattering for negative time direction is defined by a similar fashion.
SCATTERING FOR THE GENERALIZED KDV EQUATION 5
Our first result is the scattering for small data.
Theorem 1.6 (Small data scattering).Let 8/5< α < 2. Then there exists
ε0>0such that if u0H1H0,1(R)satisfies ku0kH1H0,16ε0, then
the unique H1H0,1-solution uto (1.1) given in Theorem 1.3 scatters in
H1H0,1for both time directions. Moreover,
kV(t)ukL(R;H1H0,1)+kukS(R)+ sup
tRhti1
3ku(t)kL
x.ku0kH1H0,1.
We remark that the scattering in ˆ
Lαand H1ˆ
Lαhold with a weaker
smallness assumption for 8/5< α < 2. More precisely, for u0ˆ
Lαif
kV(t)u0kS(R)+k|Dx|3
41
2αV(t)u0kL
20α
103α
xL
10
3
t(R)
is sufficiently small, then the unique ˆ
Lα-solution u(t) scatters in ˆ
Lαfor
both time directions. We emphasize that the smallness of ku0kˆ
Lαis a suf-
ficient condition for this assumption but not a necessary condition. By the
persistence-of-regularity argument, one sees that if u0H1in addition
then u(t) scatters in H1ˆ
Lα. Although Theorem 1.3 follows by a similar
persistence-of-regularity type argument, a stronger smallness assumption is
required in Theorem 1.6.
The second main result is the two equivalent characterizations of the
scattering in the weighted Sobolev space.
Theorem 1.7 (Scattering criterion).Assume 8/5< α < 2. Let u(t)be
a unique maximal-lifespan H1H0,1-solution of (1.1) under (1.8). The
following statements are equivalent:
(i) u(t)scatters for positive time direction in H1H0,1;
(ii) u(t)is bounded in a weighted norm, i.e., for some t0Imax,
kV(t)ukL
tH0,1
x([t0,Tmax)) <+.(1.10)
(iii) There exist κ > α
3(α1)(2α+1) and t0Imax such that
kukS([t0,Tmax)) + sup
t[t0,Tmax)htiκkukL2(2α+1)
x<+.
Further, if one of the above is satisfied then Tmax =and
kV(t)ukL([t0,);H1H0,1)+kukS([t0,)) + sup
t[t0,)hti1
3ku(t)kL
x<
for any t0Imax. The similar statements are true for negative time direc-
tion.
Remark 1.8.For a ˆ
Lα-solution, the boundedness kukS([t0,Tmax)) <is a
necessary and sufficient condition for scattering in ˆ
Lαfor positive time di-
rection. The equivalence of (i) and (iii) in Theorem 1.7 implies that the
additional boundedness condition
sup
t[t0,Tmax)htiκkukL2(2α+1)
x<+
bridges the gap between scattering in ˆ
Lαand in H1H0,1. This gap arises
due to the weakness of our persistence result. A standard persistence-of-
regularity argument shows that kukS([t0,Tmax)) <is also an equivalent
6 SATOSHI MASAKI AND JUN-ICHI SEGATA
characterization of scattering for positive time direction in H1ˆ
Lαfor H1
ˆ
Lα-solutions.
We remark that the implication “(ii)(i)” in Theorem 1.7 reads as a
conditional scattering result. Indeed, it establishes the scattering under the
hypothesis of the a priori bound (1.10). As mentioned above, Kim [24]
showed a conditional scattering result for 5/3< α < 2 under the bound-
edness in H1and in a Fourier-Bourgain-Morrey space. Compared with the
result, Theorem 1.7 covers a wider range 8/5< α < 2 with a stronger
boundedness assumption.
Let us compare the conditional scattering result Theorem 1.7 with the
similar results for the mass-subcritical nonlinear Schr¨odinger equation:
i∂tu+ u=µ|u|2αu, t R, x Rd,
u(0, x) = u0(x), x Rd,
(1.11)
where u:R×RdCis an unknown function, u0:RdCis a given
function, and µR\{0}and 0 < α < 2/d are constants. For (1.11) with
the defocusing nonlinearity (i.e., µ > 0), by utilizing the pseudo-conformal
transform or pseudo-conformal conservation law, it is shown in [39, 16, 1,
33] that any H0,1-solution scatters in H0,1when α>αSt := (d+ 2 +
d2+ 12d+ 4)/(4d). As far as the authors know, this kind of transform or
conservation law are not known for (1.1). As for the conditional scattering,
Killip, Murphy, Vi¸san and the first author [22, 23] proved scattering under
the boundedness assumption with respect to a scaling critical homogeneous
weighted norm or to a homogeneous Sobolev norm (see [28, 29] for similar
study for µ < 0).
1.3. Outline of the proof. To investigate the property V(t)u(t)H0,1,
it is convenient to introduce the operator
J(t) := V(t)xV (t) = x3t∂2
x.
One strategy is that, we establish a persistence-type property in the weighted
Sobolev space by looking at the equation for Ju. This argument works well
for the NLS equation (1.11). However, for the generalized KdV equation
(1.1), the operator J(t) does not work well with the nonlinear term. To
overcome this difficulty, as in Hayashi and Naumkin [12, 13, 14], we introduce
another variable
v(t) := J(t)u(t) + 3µt|u(t)|2αu(t).(1.12)
Note that if u(t) is a solution to (1.1) then one has v(t) = (x+ 31
xt)u,
at least formally. We would like to point out that our vdoes not involve an
anti-derivative 1
x. A direct computation shows that vsolves a KdV-like
equation
(1.13) tv+3
xv= (2α+ 1)µ|u|2αxv2(α1)µ|u|2αu.
It is noteworthy that the equation is written in the integral form and hence
that one can utilize the Strichartz estimates to obtain various estimates for
v.
The following notation will be used throughout this paper: We use A.B
to denote the estimate A6CB where Cis a positive constant. |Dx|s=
SCATTERING FOR THE GENERALIZED KDV EQUATION 7
(2
x)s/2and hDxis= (I2
x)s/2denote the Riesz and Bessel potentials
of order s, respectively. For 1 6p, q 6and IR, let us define a
space-time Lebesgue spaces
Lq
tLp
x(I) = {u:I×RR;kukLq
tLp
x(I)<∞},
kukLq
tLp
x(I)=kku(t, ·)kLp
x(R)kLq
t(I),
Lp
xLq
t(I) = {u:I×RR;kukLp
xLq
t(I)<∞},
kukLp
xLq
t(I)=kku(·, x)kLq
t(I)kLp
x(R).
The rest of the paper is organized as follows. In Section 2, we review the
well-posedness theory for (1.1) in the Fourier-Lebesgue space. Sections 3 is
devoted to the proof of Theorems 1.3 and 1.4. In Sections 4 and 5, we prove
Theorems 1.6 and 1.7, respectively.
2. Well-posedness in the Fourier-Lebesgue space
In this section, we review the well-posedness theory for (1.1) in the Fourier-
Lebesgue space ˆ
Lα. Furthermore, we prove the long time perturbation for
(1.1) in the Fourier-Lebesgue space.
We first review the space-time estimates in ˆ
Lαof solution to the Airy
equation tu+3
xu=F(t, x), t I, x R,
u(0, x) = f(x), x R,
(2.1)
where IRis an interval, F:I×RRand f:RRare given functions.
Let {V(t)}tRbe an unitary group in L2defined by
(V(t)f)(x) = 1
2πZ
−∞
eixξ+itξ3ˆ
f(ξ)dξ.
Using the group, the solution to (2.1) can be written as
u(t) = V(t)f+Zt
0
V(tτ)F(τ)dτ.
Proposition 2.1 (homogeneous space-time estimates).Let Ibe an interval.
Let (p, q)satisfy
061
p<1
4,061
q<1
21
p.
Then, for any fˆ
Lr,
(2.2) k|Dx|sV(t)fkLp
xLq
t(I)6Ckfkˆ
Lr,
where 1
r=2
p+1
q, s =1
p+2
q
and positive constant Cdepends only on rand s.
Proof of Proposition 2.1. For the proof of (2.2) with (p, q , r) = (4,,2) or
(p, q, r) = (,2,2), see [19, Theorem 2.5] and [19, Theorem 4.1], respec-
tively. For the proof of (2.2) with p=qand r > 4/3, see Gr¨unrock [10,
Corollary 3.6] or [30, Lemma 2.2]. The general case follows from the above
cases and the interpolation. See [30, Proposition 2.1] for the detail.
8 SATOSHI MASAKI AND JUN-ICHI SEGATA
Proposition 2.2 (inhomogeneous space-time estimates).Let 4/3< r < 4
and let (pj, qj)(j= 1,2) satisfy
061
pj
<1
4,061
qj
<1
21
pj
.
Then, the inequalities
(2.3)
Zt
0
V(tτ)F(τ)
L
t(I;ˆ
Lr
x)
6C1k|Dx|s2FkLp
2
xLq
2
t(I),
and
(2.4)
|Dx|s1Zt
0
V(tτ)F(τ)
Lp1
xLq1
t(I)
6C2k|Dx|s2FkLp
2
xLq
2
t(I)
hold for any Fsatisfying |Dx|s2FLp
2
xLq
2
twith
1
r=2
p1
+1
q1
, s1=1
p1
+2
q1
and
1
r=2
p2
+1
q2
, s2=1
p2
+2
q2
,
where the constant C1depends on r,s2and I, and the constant C2depends
on r,s1,s2and I.
Proof of Proposition 2.2. (2.3) and (2.4) follow from Proposition 2.1 and
Christ-Kiselev lemma [2] (see also [17, Lemma 2.5] for the space-time norm
version of Christ-Kiselev lemma). See [30, Proposition 2.5] for the detail.
Next, we review the small data scattering in ˆ
Lαfor (1.1) obtained by [30].
Lemma 2.3. Let 8/5< α < 2. Then there exists ˜ε > 0such that if
u0ˆ
Lα
x(R)satisfies ku0kˆ
Lα
x6˜ε, then there exists a global ˆ
Lα-solution uto
(1.1) satisfying
kukL
t(R;ˆ
Lα
x)+kukS(R)62ku0kˆ
Lα
x.(2.5)
Further, u(t)scatters in ˆ
Lαfor both time directions.
Proof of Lemma 2.3. See [30, Theorem 1.7].
Next we prove the long time perturbation lemma for (1.1) in the Fourier-
Lebesgue space. We define
X(I) := {u:I×RR;kukX(I)<∞},
kukX(I):= k|Dx|sukL
20α
103α
xL
10
3
t(I)
,
Y(I) := {u:I×RR;kukY(I)<∞},
kukY(I):= k|Dx|sukL
20α
10+13α
xL
10
7
t(I)
with s=3
41
2α.
SCATTERING FOR THE GENERALIZED KDV EQUATION 9
Proposition 2.4 (Long time perturbation).Assume 8/5< α < 2. For
any M > 0there exists ε > 0such that the following property holds: Let
t0Rand let IRbe an interval such that t0I. Let eu:I×RR
be a function such that euS(I)X(I), where S(I)is given by (1.6). Put
E:= (t+3
x)euµ∂x(|eu|2αeu). Let u0ˆ
Lα. Suppose that
keukS(I)X(I)6M(2.6)
and
V(tt0)(u0eu(t0)) Zt
t0
V(tτ)E(τ)
S(I)X(I)
6ε.(2.7)
Then, the unique ˆ
Lα-solution u(t)of (1.1) satisfying u(t0) = u0exists on I
and satisfies
kueukS(I)X(I).Mε.
To prove Proposition 2.4, we use the Leibniz rule for the fractional deriva-
tives obtained by [3] and [20].
Lemma 2.5. Assume β(0,1). Let p, p1, p2, q, q2(1,)and q1(1,]
satisfy 1/p = 1/p1+ 1/p2and 1/q = 1/q1+ 1/q2. We also assume F
C1(R;R). Then for any interval I, the inequality
(2.8) k|Dx|βF(f)kLp
xLq
t(I).kF(f)kLp1
xLq1
t(I)k|Dx|βfkLp2
xLq2
t(I)
holds for any fsatisfying F(f)Lp1
xLq1
t(I)and |Dx|βfLp2
xLq2
t(I), where
the implicit constant depends only on β, p1, p2, q1, q2and I.
Proof of Lemma 2.5. See [3, Proposition 3.1] and [20, Theorem A.6]. Note
that the alternative proof of the inequality (2.8) can be found in [30, Lemma
3.7].
Lemma 2.6. Let β(0,1), β1, β2[0, β]satisfy β=β1+β2and let
p, p1, p2, q, q1, q2(1,)satisfy 1/p = 1/p1+ 1/p2and 1/q = 1/q1+ 1/q2.
Then for all interval I, the inequality
k|Dx|β(fg)f|Dx|βgg|Dx|βfkLp
xLq
t(I)
6Ck|Dx|β1fkLp1
xLq1
t(I)k|Dx|β2gkLp2
xLq2
t(I)
holds for any fand gsatisfying |Dx|β1fLp1
xLq1
t(I)and |Dx|β2gLp2
xLq2
t(I),
where the implicit constant depends only on β1, β2, p1, p2, q1, q2and I.
Proof of Lemma 2.6. See [20, Theorem A.8].
Proof of Proposition 2.4. It suffices to consider the case inf I=t0. The
general case follows by splitting I= (I[t0,))(I(−∞, t0]) and applying
the time reversal symmetry to estimate the latter. Further, we may let t0= 0
without loss of generality by the time translation symmetry.
By the assumption (2.6), we see that for any η > 0 there exist N=
N(M, η) and a subdivision {tj}N
j=0 of [0,) with 0 = t0< t1<···< tN=
+such that
keukS(Ij)+keukX(Ij)< η
holds for all i[1, N ], where Ij:= [tj1, tj).
10 SATOSHI MASAKI AND JUN-ICHI SEGATA
Let us first consider the equation for w:= ueuon I1= [0, t1):
w(t) = µZt
0
V(tτ)x(|w+eu|2α(w+eu) |eu|2αeu)(2.9)
+N(t),
where
N(t) := V(t)(u0eu(0)) Zt
0
V(tτ)E(τ)dτ.
By Proposition 2.2 and Lemma 2.5, we obtain
kwkS(I1)X(I1)6kNkS(I1)X(I1)
+C(kwkX(I1)+keukX(I1))(kwk2α1
S(I1)+keuk2α1
S(I1))kwkS(I1)
+C(kwk2α
S(I1)+keuk2α
S(I1))kwkX(I1)
6ε+C(kwkX(I1)+η)(kwk2α1
S(I1)+η2α1)kwkS(I1)
+C(kwk2α
S(I1)+η2α)kwkX(I1)
6ε+Cη2αkwkS(I1)X(I1)+Ckwk2α+1
S(I1)X(I1).
We remark that Ccan be chosen independently of M,η, and ε. If ηis small
then this implies
kwkS(I1)X(I1)62ε+ 2Ckwk2α+1
S(I1)X(I1).
There exists a constant δ > 0 such that if 2ε6δthen this implies that
kwkS(I1)X(I1)64ε.
Now, let j[2, N] and suppose that we can choose εj1so that if ε6εj1
then
kwkS(Ik)X(Ik)64kε6η
holds for k[1, j 1]. Let us next consider the equation (2.9) for won
Ij= [tj1, tj). We rewrite (2.9) as
w(t) = µZtj1
0
V(tτ)x(|w+eu|2α(w+eu) |eu|2αeu)
+µZt
tj1
V(tτ)x(|w+eu|2α(w+eu) |eu|2αeu) +N(t).
SCATTERING FOR THE GENERALIZED KDV EQUATION 11
By Proposition 2.2, one has
Ztj1
0
V(tτ)x(|w+eu|2α(w+eu) |eu|2αeu)
S(Ij)X(Ij)
=
Zt
0
V(tτ)1[0,tj1)(τ)x(|w+eu|2α(w+eu) |eu|2αeu)
S(Ij)X(Ij)
6
Zt
0
V(tτ)1[0,tj1)(τ)x(|w+eu|2α(w+eu) |eu|2αeu)
S([0,tj))X([0,tj))
.k1[0,tj1)x(|w+eu|2α(w+eu) |eu|2αeu)kY([0,tj))
=kx(|w+eu|2α(w+eu) |eu|2αeu)kY([0,tj1))
6
j1
X
k=1 kx(|w+eu|2α(w+eu) |eu|2αeu)kY(Ik)
6
j1
X
k=1
2Cη2α4kε68
3Cη2α4j1ε.
Hence,
kwkS(Ij)X(Ij)6kNkS(Ij)X(Ij)+8
3Cη2α4j1ε
+C(kwkX(Ij)+keukX(Ij))(kwk2α1
S(Ij)+keuk2α1
S(Ij))kwkS(Ij)
+C(kwk2α
S(Ij)+keuk2α
S(Ij))kwkX(Ij)
6ε+8
3Cη2α4j1ε+Cη2αkwkS(Ij)X(Ij)+Ckwk2α+1
S(Ij)X(Ij).
Letting ηeven smaller if necessary, we have 2α61
4and hence
kwkS(Ij)X(Ij)64
3(1 + 2
34j1)ε+ 2Ckwk2α+1
S(Ij)X(Ij).
Hence, if ε6min(2
3(1 + 2
34j1)1δ, 4jη, εj1) =: εjthen
kwkS(Ij)X(Ij)68
3(1 + 2
34j1)ε64jε6η.
Hence, by induction, we can choose εNsuch that if ε6εNthen
kwkS(Ij)X(Ij)64jε6η
holds for j[1, N ]. Combining this estimate and noting that Ndepends
on M, we obtain
kwkS(I)X(I).Mε.
In the end of this section, we prove the compactness of the embedding
H1H0,1֒ˆ
Lα.
Lemma 2.7. The embedding H1H0,1֒ˆ
Lαis compact for 16α6.
Proof of Lemma 2.7. It is an immediate consequence of the embedding H3/4
H0,3/4֒L1Lholds and the fact that the embedding H1H0,1֒
H3/4H0,3/4is compact.
12 SATOSHI MASAKI AND JUN-ICHI SEGATA
3. Proof of Theorems 1.3 and 1.4
In this section, we prove local well-posedenss and blowup alternative. Fix
t0and u0H1such that J(t0)u0L2. Note that u0H1ˆ
Lα. Indeed,
(3.1)
kFu0kLα=kFV(t0)u0kLα
.kFV(t0)u0k
3α2
2α
L2kξFV(t0)u0k
2α
2α
L2
=ku0k
3α2
2α
L2kJ(t0)u0k
2α
2α
L2<.
Hence, by the local well-posedness result in ˆ
LαH1[30, Theorem 1.5], one
obtains a ˆ
LαH1-solution uto (1.1) in a neighborhood Iof t0. In particular,
one has
kukL
t(I;H1
x(R)) +
2
X
k=1 kk
xukL
x(R;L2
t(I)) .ku0kH1.
We note that the size of the neighborhood is chosen so that kV(tt0)u0kS(I)
is smaller than a universal constant. Hence, what we have to do is to show
that the H1ˆ
Lα-solution uis a H1H0,1-solution. To this end, we estimate
Ju by considering
v:= Ju + 3µt|u|2αu
defined in (1.12). We further introduce
P:= x∂x+ 3t∂t.
We have the identity
(3.2) xv=P u +u.
Before the proof, let us derive an equation for Ju and P u. We also confirm
that vsolves (1.13). Let L=t+3
x. Suppose that uC(I;H1) solves
Lu =µ∂x(|u|2αu)
in the distribution sense. Let us note beforehand that the following calcu-
lation is valid in the distribution sense. Operating Jto the both sides and
noting [L, J ] = 0, we see
LJu =µJx(|u|2αu).
It holds that
Jx=P3tL.(3.3)
Hence, we have
LJu =µP (|u|2αu)3µtL(|u|2αu)(3.4)
= (2α+ 1)µ|u|2αP u 3µtL(|u|2αu).
Since [J, x] = 1, another use of the above identity yields
P u =J∂xu+ 3tLu =xJ u u+ 3µt∂x(|u|2αu)(3.5)
=xvu.
This is (3.2). Furthermore, since [L, t] = 1, we have
tL(|u|2αu) = Lt(|u|2αu) |u|2αu.(3.6)
SCATTERING FOR THE GENERALIZED KDV EQUATION 13
Substituting (3.5) and (3.6) into (3.4), we obtain
Lv = (2α+ 1)µ|u|2αxv2(α1)µ|u|2αu,
which is nothing but (1.13). On the other hand, if we operate Pto the
equation for u, we obtain
P Lu =µP x(|u|2αu).
Using the relations [P, L] = 3Land [P, x] = x, we obtain
LP u = 3Lu +µ∂xP(|u|2αu)µ∂x(|u|2αu)(3.7)
= (2α+ 1)µ∂x(|u|2αP u) + 2µ∂x(|u|2αu).
Thus, we see from (3.4) that
(3.8) t(Ju) + 3
x(Ju) = (2α+ 1)µ|u|2αP u 3µt(t+3
x)(|u|2αu).
Further, by (3.7),
(3.9) t(P u) + 3
x(P u) = (2α+ 1)µ∂x(|u|2αP u) + 2µ∂x(|u|2αu).
The local well-posedness in the weighted Sobolev space H1H0,1(The-
orem 1.3) is a consequence of the following persistence-type result.
Lemma 3.1. Let t0Rand let u0ˆ
LαH1. Let u(t)be a ˆ
LαH1-
solution to (1.1) under (1.8). There exists a constant δ > 0such that if
V(t0)u0H1H0,1then u(t)is a H1H0,1-solution to (1.1) on any
interval It0satisfying kukS(I)6δ. Further,
kJukL
tL2
x(I)+kvkL
tL2
x(I)+kxvkL
xL2
t(I)
.kV(t0)u0kH1
x+ht0iku0k2α+1
H1
x,
where vis defined by (1.12).
Proof of Lemma 3.1. Let us prove that the H1ˆ
Lα-solution satisfies the
desired weighted estimate. To this end, we obtain an estimate of vdefined
in (1.12) by solving (1.13) under the initial condition
v(t0) = v0:= J(t0)u0+ 3µt0|u0|2αu0L2.
For R > 0 and T > 0, we define a complete metric space
ZR,T := {vC(IT;L2
x) ; kvkZ(IT)6R}
with the distance
d(v1, v2) = kv1v2kZ(IT),
where IT= (t0T , t0+T),
(3.10) kvkZ(I):= kvkL
tL2
x(I)+kxvkL
xL2
t(I).
We suppose that T > 0 is small so that ITI. Let us prove that the map
Φ(v) defined by
Φ(v)(t) := V(tt0)v0+ (2α+ 1)µZt
t0
V(tτ)(|u|2αxv)(τ)
2(α1)µZt
t0
V(tτ)(|u|2αu)(τ)
is a contraction map from ZR,T to itself.
14 SATOSHI MASAKI AND JUN-ICHI SEGATA
Pick vZR,T . By Propositions 2.1 and 2.2,
kΦ(v)kZ(IT)6Ckv0kL2
x+Ck|u|2αxvkL
5
4
xL
10
9
t(IT)+Ck|u|2αukL1
tL2
x(IT)
6Ckv0kL2
x+Ckuk2α
S(IT)kxvkL
xL2
t(IT)+CT kuk2α+1
L
tH1
x(IT)
6Ckv0kL2
x+Ckuk2α
S(IT)R+CT ku0k2α+1
H1
x.
We first choose T61 so small that Ckuk2α
S(IT)61
2and then we let
R= 2C(kv0kL2
x+ku0k2α+1
H1
x).
Then, one sees that Φ is a map from ZR,T to itself. Similarly, for v1, v2
ZR,T , one obtains
Φ(v1)Φ(v2) = (2α+ 1)µZt
t0
V(tτ)(|u|2αx(v1v2))(τ)
and hence, estimating as above, one sees that
d(Φ(v1),Φ(v2)) 6Ckuk2α
S(IT)kx(v1v2)kL
xL2
t(IT)61
2d(v1, v2),
which shows that Φ is a contraction map. Thus, we see that vC(IT;L2
x)
obeys the bound
kvkZ(IT)6R.kJ(t0)u0kL2
x+ht0iku0k2α+1
H1
x.
So far, we construct vas a solution to (1.13). Let us prove that v=
Ju + 3µt|u|2αuholds in the distribution sense, which implies that J(t)u(t)
C(IT;L2
x) and
kJukL
tL2
x(IT).kvkZ(IT)+ht0ikuk2α+1
L(IT;H1).kJ(t0)u0kL2+ht0iku0k2α+1
H1
x.
To this end, we put
z=xvu, w =v3µt|u|2αu.
By (1.13), one obtains
(t+3
x)z=x((2α+ 1)µ|u|2αxv2(α1)µ|u|2αu)µ∂x(|u|2αu)
= (2α+ 1)µ∂x(|u|2α(z+u)) (2α1)µ∂x(|u|2αu)
= (2α+ 1)µ∂x(|u|2αz) + 2µ∂x(|u|2αu).
Hence, zsolves (3.9) in the distribution sense. Together with
z(t0) = xv(t0)u(t0) = x∂xu0+ 3t0(3
xu0+µ∂x(|u0|2αu0)) = (P u)(t0),
we see that z=P u. Hence, we further obtain
tw+3
xw= (t+3
x)v3µ|u|2αu3µt(t+3
x)(|u|2αu)
= (2α+ 1)µ|u|2αxv(2α+ 1)µ|u|2αu3µt(t+3
x)(|u|2αu)
= (2α+ 1)µ|u|2αP u 3µt(t+3
x)(|u|2αu),
i.e., wsolves (3.8) in the distribution sense. Since
w(t0) = v(t0)3µt0|u0|2αu0=J(t0)u0,
we see that w=Ju. Thus, v=Ju + 3µt|u|2αuholds.
We conclude this section with the proof of Theorem 1.4.
SCATTERING FOR THE GENERALIZED KDV EQUATION 15
Proof of Theorem 1.4. Let u(t) be a maximal-lifespan H1ˆ
Lα-solution given
in [30, Theorem 1.9]. Here, the maximal lifespan Imax = (Tmin, Tmax) is
that as a H1ˆ
Lα-solution. Let us prove that this is also a maximal-lifespan
in the sense of H1H0,1-solution. Recall that Tmax <implies
kukS([t0,Tmax)) =
(see [30, Theorem 1.5]). Hence, it suffices to prove that, for any finite T > t0,
kukS([t0,T )) <= kJukL
tL2
x([t0,T )) <.
Let δ > 0 be the number given in Lemma 3.1. We can obtain a subdivision
{tj}N
j=1 of [t0, T ):
t0< t1< t2<···< tN=T
so that N.α,kukS([t0,T )) 1 and kukS([tj1,tj)) 6δfor all j[1, N ]. By apply-
ing Lemma 3.1 to each interval [tj1, tj), we obtain kJukL
tL2
x([tj1,tj)) <
for all j[1, N ]. This implies the desired boundedness kJukL
tL2
x([t0,T )) <
. This completes the proof.
4. Proof of Theorem 1.6
To prove Theorem 1.6, we employ the well-posedness result of (1.1) in the
Fourier-Lebesgue space ˆ
Lα(R) mentioned in Section 2.
Lemma 4.1. Let 8/5< α < 2. Let t0Rand suppose that V(t0)u0
H1H0,1. There exists ε1>0such that if ε=kV(t0)u0kH1H0,16ε1
then the H1H0,1-solution to (1.1) under (1.8) is global and satisfies
kukS(R).ε.(4.1)
Proof of Lemma 4.1. By (3.1), we see that ku0kˆ
Lα.ε. Hence Lemma 2.3
yields that if εis sufficiently small, then there exists a global ˆ
Lα-solution u
satisfying (4.1). By Theorem 1.4, uis a global H1H0,1-solution.
Lemma 4.2. Let 8/5< α < 2. Let t0Rand suppose that V(t0)u0
H1H0,1. Let ube the unique maximal-lifespan H1H0,1-solution to (1.1)
under (1.8). Then there exists δ2>0such that if an interval It0satisfies
IImax and
kukS(I)6δ2
then it holds that
(4.2) kukL
tH1
x(I)+kxukL
xL2
t(I)+k2
xukL
xL2
t(I).ku0kH1
x.
In particular, there exists ε2(0, ε1]such that if ε=kV(t0)u0kH1H0,16
ε2, then the solution is global and satisfies (4.1) and
kukL
tH1
x(R)+kxukL
xL2
t(R)+k2
xukL
xL2
t(R).ε,(4.3)
where ε1is the number given in Lemma 4.1.
Proof of Lemma 4.2. The latter half follows from the former half and the
previous lemma. Hence, let us prove the former part. We omit (I) in the
norm, for simplicity.
16 SATOSHI MASAKI AND JUN-ICHI SEGATA
By Propositions 2.1 and 2.2, we have
kukL
tH1
x+kxukL
xL2
t+k2
xukL
xL2
t
(4.4)
.ku0kH1
x+kx(|u|2αu)kL
5
4
xL
10
9
t
+k2
x(|u|2αu)kL
5
4
xL
10
9
t
.ku0kH1
x+kuk2α
SkxukL
xL2
t+kuk2α1
Skxuk2
L5α
xL
20α
5α+2
t
+kuk2α
Sk2
xukL
xL2
t.
Since
kxukL5α
xL
20α
5α+2
t
.kuk
1
2
Sk2
xuk
1
2
L
xL2
t,
substituting this and Lemma 4.1 into (4.4), we obtain
kukL
tH1
x+kxukL
xL2
t+k2
xukL
xL2
t
.ku0kH1
x+kuk2α
S(kxukL
xL2
t+k2
xukL
xL2
t)
.ku0kH1
x+δ2α
2(kxukL
xL2
t+k2
xukL
xL2
t).
Hence if δ2is sufficiently small, then we have the desired estimate.
Corollary 4.3. Let 8/5< α < 2. Let t0Rand suppose that V(t0)u0
H1H0,1. Let ube the unique maximal-lifespan H1H0,1-solution to (1.1)
under (1.8). If kukS(I)<holds for an interval Ithen we have
kukL
tH1
x(I)+kxukL
xL2
t(I)+k2
xukL
xL2
t(I)<.
Proof of Corollary 4.3. We subdivide the interval Iso that S-norm of the
solution on each subinterval is smaller than the constant δ2in Lemma 4.2.
Note that the number of the subinterval depends only on αand kukS(I).
Then, a recursive use of Lemma 4.2 yields the result.
Now, let us turn to the global bound on Ju.
Lemma 4.4. Let 8/5< α < 2. Let t0Rand suppose that V(t0)u0
H1H0,1. There exists ε3(0, ε2]such that if ε=kV(t0)u0kH1H0,16ε3,
then the unique global H1H0,1-solution to (1.1) under (1.8) satisfies (4.1),
(4.3), and
(4.5) sup
tRhti1
3ku(t)kL
x+kJukL
tL2
x(R)+kvkL
tL2
x(R)+kxvkL
xL2
t(R).ε,
where ε2is the number given in Lemma 4.2.
To prove Lemma 4.4, we show the Klainerman-Sobolev type inequality.
Lemma 4.5 (Klainerman-Sobolev type inequality).Let t6= 0 and p
[2,]. For any uL2
xsatisfying J(t)uL2
x, we have
kukLp
x.|t|1
3+2
3pkuk
1
2+1
p
L2
xkJuk
1
21
p
L2
x.
Proof of Lemma 4.5. We consider the case p=. By the elementary prop-
erty of the Airy function, we see
kV(t)fkL
x.t1
3kfkL1
x.
SCATTERING FOR THE GENERALIZED KDV EQUATION 17
Hence by the L2unitary property of the group V(t),
ku(t)kL
x=kV(t)V(t)ukL
x
.t1
3kV(t)ukL1
x
.t1
3kV(t)uk
1
2
L2
xkxV (t)uk
1
2
L2
x
=Ct1
3kuk
1
2
L2
xkJ(t)uk
1
2
L2
x.
Hence, we obtain the L-estimate. Note that L2-estimate is obvious by the
unitary property of V(t). The general case follows by interpolation.
Proof of Lemma 4.4. Suppose that ε6ε2. Then, the global solution u(t)
satisfies (4.1) and (4.3). We prove the bound (4.5) on [0,).
By the definition of v,
kJukL2
x.kvkL2
x+tk|u|2αukL2
x.
By the Sobolev and the Kleinerman-Sobolev inequalities (Lemma 4.5),
ku(t)kL
x.(kukH1
x.εfor 0 6t61,
t1
3kuk
1
2
L2
xkJuk
1
2
L2
x.ε1
2t1
3kJuk
1
2
L2
xfor t>1.
Hence
k|u|2αukL2
x.1[0,1](t)ε2α+1 +1[1,](t)εα+1t2
3αkJukα
L2
x,(4.6)
where 1Ais a characteristic function on the set A. Therefore, for any T > 1
kJukL
tL2
x(IT).kvkL
tL2
x(IT)+ε2α+1 +εα+1kJukα
L
tL2
x(IT),(4.7)
where IT= [0, T ). By Propositions 2.1 and 2.2, (4.1), and (4.6),
kvkL
tL2
x(IT)+kxvkL
xL2
t(IT)
(4.8)
.kxu0kL2
x+k|u|2αxvkL
5
4
xL
10
9
t(IT)+k|u|2αukL1
tL2
x(IT)
.kxu0kL2
x+kuk2α
S(IT)kxvkL
xL2
t(IT)+k|u|2αukL1
tL2
x(IT)
.ε+ε2αkxvkL
xL2
t(IT)+ε2α+1 +εα+1kJukα
L
tL2
x(IT).
By (4.7) and (4.8),
kJukL
tL2
x(IT)+kvkL
tL2
x(IT)+kxvkL
xL2
t(IT)
.ε+ε2αkxvkL
xL2
t(IT)+ε2α+1 +εα+1kJukα
L
tL2
x(IT).
Hence letting kukAT:= kJukL
tL2
x(IT)+kvkL
tL2
x(IT)+kxvkL
xL2
t(IT), we
have
kukAT.ε+ε2αkukAT+kukα
AT.
Hence if εis sufficiently small, then this inequality implies that kukAT.ε.
Since T > 1 is arbitrary, we have kukA.ε. Finally, combining kukA.ε
and Lemma 4.5, we have
sup
tRhti1
3ku(t)kL.ε.
This completes the proof of (4.5).
We now turn to the scattering in H1H0,1.
18 SATOSHI MASAKI AND JUN-ICHI SEGATA
Lemma 4.6. Let 8/5< α < 2. Let ube a maximal-lifespan H1H0,1-
solution to (1.1). Pick t0Imax . If
kukS([t0,Tmax)) +kJukL
tL2
x([t0,Tmax)) <,
then u(t)scatters in H1H0,1for positive time direction. A similar state-
ment holds for the negative time direction.
Proof of Lemma 4.6. By the blowup criteria, we have Tmax =. Hence,
without loss of generality, we may suppose that t0>0. By Corollary 4.3,
we have
kukL
tH1
x([t0,)) +kxukL
xL2
t([t0,)) +k2
xukL
xL2
t([t0,)) <.
We shall show that V(t)u(t) is a Cauchy sequence in H1H0,1. As in
the proof of Lemma 4.2, for t06s < t, we have
kV(t)u(t)V(s)u(s)kH1
x
=|µ|
Zt
s
V(τ)x(|u|2αu)
H1
x
.kx(|u|2αu)kL
5
4
xL
10
9
t(s,t)+k2
x(|u|2αu)kL
5
4
xL
10
9
t((s,t))
.kuk2α
S((s,t))(kxukL
xL2
t([t0,)) +k2
xukL
xL2
t([t0,)))
0 as s .
Let us turn to the estimate in H0,1. By (1.12),
kx(V(t)u(t)V(s)u(s))kL2
x
(4.9)
=kV(t)J(t)u(t)V(s)J(s)u(s)kL2
x
.kV(t)v(t)V(s)v(s)kL2
x
+tk|u|2αu(t)kL2
x+sk|u|2αu(s)kL2
x.
By assumption and Lemma 4.5, we have ku(t)kL=O(t1/3). Hence,
together with the mass conservation (1.3), one sees that the last two terms
in the right hand side of (4.9) vanish as s, t . Further, since vsatisfies
(1.13), Proposition 2.2 yields
kV(t)v(t)V(s)v(s)kL2
x
(4.10)
.
Zt
s
V(τ)|u|2αxvdτ
L2
x
+
Zt
s
V(τ)|u|2αudτ
L2
x
.k|u|2αxvkL
5
4
xL
10
9
t((s,t)) +k|u|2αukL1
tL2
x((s,t))
.kuk2α
S(s,t)kxvkL
xL2
t([t0,)) +s2
3α+1(sup
t>1
t1
3ku(t)kL)2αku0kL2
0 as s .
Plugging (4.10) to (4.9), we obtain
kx(V(t)u(t)V(s)u(s))kL2
x0 as s .
Therefore we have that V(t)u(t) is a Cauchy sequence in H1H0,1. This
implies that u(t) scatters in H1H0,1for positive time direction.
SCATTERING FOR THE GENERALIZED KDV EQUATION 19
Proof of Theorem 1.6. Theorem 1.6 is an immediate consequence of Lemmas
4.4 and 4.6.
5. Proof of Theorem 1.7
In this section we prove Theorem 1.7.
Proof of Theorem 1.7. Let u(t) be a maximal-lifespan H1H0,1-solution.
Step 1. Let us prove “(iii)(ii)”. Suppose that for some κ > α
3(α1)(2α+1)
and t0Imax,
R:= kukS([t0,Tmax)) + sup
t[t0,Tmax)htiκku(t)kL2(2α+1)
x<.
By Theorem 1.4, we see that Tmax =. Further, by Corollary 4.3, we
obtain
kukL((t0,);H1
x)<.
We claim that there exists ˜κ > 1
2α+1 such that
(5.1) sup
t>t1hti˜κku(t)kL2(2α+1)
x<
for some t1>t0. We consider the case κ61
2α+1 since if κ > 1
2α+1 then this
is trivial by choosing ˜κ=κ. Let us consider the case κ < 1
2α+1 . Let δ0>0
be a constant to be determined later. For any choice of δ0>0, there exists
t1Imax [max(t0,1),) such that
kukS((t1,)) 6δ0.
We apply Propositions 2.1 and 2.2, and the assumption to obtain
kvkZ((t1,T )) 6Ckv(t1)kL2+Ckuk2α
S((t1,T ))kvkZ((t1,T )) +k|u|2αukL1
tL2
x((t1,T ))
6Ckv(t1)kL2+Ckuk2α
S((t1,T ))kvkZ((t1,T )) +kt(2α+1)κkL1
t(t1,T )R2α+1
6Ckv(t1)kL2+2α
0kvkZ((t1,T )) +CR2α+1 T1(2α+1)κ,
where Z(I) is as in (3.10). We choose δ0so that Cδ2α
061
2. Then, we see
that
kvkZ((t1,T )) 62Ckv(t1)kL2+ 2CR2α+1T1(2α+1)κ
for any T(t1,). In particular, we obtain
(5.2) kv(t)kL2.t1(2α+1)κ
for all t>t1(>1). One then sees from this inequality, Lemma 4.5, and the
mass conservation (1.3) that
t2α
3(2α+1) ku(t)kL2(2α+1) 6ku0k
α+1
2α+1
L2(kv(t)kL2+tku(t)k2α+1
L2(2α+1) )α
2α+1 .tα
2α+1 ακ.
Hence,
(5.3) tκ1ku(t)kL2(2α+1)
x.1
for all t>t1, where κ1:= ακ α
3(2α+1) . One sees that
κ1κ= (α1) κα
3(α1)(2α+ 1)>0