Smoothing properties for the higher order nonlinear Schr\"{o}dinger
  equation with constant coefficients
Smoothing properties for the higher order nonlinear
Schrödinger equation with constant coefficients
Mauricio Sepúlveda ∗ Octavio Vera Villagrán.†
Abstract
We study local and global existence and smoothing properties for the initial value problem associated
to a higher order nonlinear Schrödinger equation with constant coefficients which appears as a model
for propagation of pulse in optical fiber.
Keywords and phrases: Evolution equations, weighted Sobolev space, gain in regularity.
Mathematics Subject Classification: 35Q53, 47J35
1 Introduction
We consider the initial value problem
i ut + ω uxx + i β uxxx + |u|
2 u = 0 x, t ∈ R
u(x, 0) = u0(x)
where ω, β ∈ R, β 6= 0 and u = u(x, t) is a complex valued function. The above equation is a particular
case of the equation
i ut + ω uxx + i β uxxx + γ |u|
2 u+ i δ |u|2 ux + i ǫ u
2 ux = 0 x, t ∈ R
u(x, 0) = u0(x)
where ω, β, γ, δ are real numbers with β 6= 0. This equation was first proposed by A. Hasegawa and Y.
Kodama [13] as a model for the propagation of a signal in an optic fiber (see also [20]). The equation
(Q) can be reduced to other well known equations. For instance, setting ω = 1, β = δ = ǫ = 0 in (Q) we
have the semilinear Schrödinger equation, i. e.,
i ut + uxx + γ |u|
2 u = 0. (Q1)
If we let β = γ = 0 and ω = 1 in (Q), we obtain the derivative nonlinear Schrödinger equation
i ut + uxx + i δ |u|
2 ux + i ǫ u
2 ux = 0. (Q2)
Letting α = γ = ǫ = 0 in (Q), the equation that arises is the complex modified Korteweg-de Vries
equation,
i ut + i β uxxx + i δ |u|
2 ux = 0. (Q3)
The initial value problem for the equations (Q1), (Q2) and (Q3) has been extensively studied in the last
few years. See, for instance, [1, 2, 3, 5, 6, 8, 9, 17, 18, 26, 27] and references therein. In 1992, C. Laurey
[22] considered the equation (Q) and proved local well-posedness of the initial value problem associated
for data in Hs(R), s > 3/4, and global well-posedness in Hs(R), s ≥ 1. In 1997, G. Staffilani [28] for (Q)
∗Departamento de Ingenieŕıa Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile. mauricio@ing-
mat.udec.cl
†Departamento de Matemática, Universidad del B́ıo-B́ıo, Collao 1202, Casilla 5-C, Concepción, Chile. overa@ubiobio.cl
http://arxiv.org/abs/0704.1862v1
established local well-posedness for data in Hs(R), s ≥ 1/4 improving Laurey’s result. A similar result
was given in [5, 6] with w(t), β(t) real functions.
Our aim in this paper, is to study gain in regularity for the equation (P ). Specifically, we prove conditions
on (P ) for which initial data u0 possessing sufficient decay at infinity and minimal amount of regularity
will lead to a unique solution u(t) ∈ C∞(R) for 0 < t < T, where T is the existence time of the solution.
We are not considering the equation (Q) because of the technique used here, we shall see that the last
two terms in (Q) are not outstanding in the main inequality, indeed the two last terms are observed in
the last two terms in the main inequality.
In 1986, N. Hayashi et al. [13] showed that for the nonlinear Schrödinger equation (NLS): i ut +
uxx = λ |u|
p−1 u, (x, t) ∈ R × R with initial condition u(x, 0) = u0(x), x ∈ R and a certain assump-
tion on λ and p, all solutions of finite energy are smooth for t 6= 0 provided the initial functions in
H1(R)(or on L2(R)) decay sufficiently fast as |x| → ∞. The main tool is the operator J defined by
Ju = ei x
2/4 t (2 i t) ∂x(e
− i x2/4 t u) = (x + 2 i t ∂x)u which has the remarkable property that it commutes
with the operator L defined by L = (i ∂t + ∂
x), namely LJ − JL = [L, J ] = 0.
For the Korteweg-de Vries type equation (KdV), J. C. Saut and M. Temam [26] remarked that a so-
lution u cannot gain or lose regularity. They showed that if u(x, 0) = u0(x) ∈ H
s(R) for s ≥ 2, then
u( · , t) ∈ Hs(R) for all t > 0. For the KdV equation on the line, Kato [17] motivated by work of Cohen
[11] showed that if u(x, 0) = u0(x) ∈ L
b ≡ H
2(R) ∩ L2(ebx dx)(b > 0) then the solution u(x, t) of the
KdV equation becomes C∞ for all t > 0. A main ingredient in the proof was the fact that formally the
semi-group S(t) = e−∂
x in L2b(R) is equivalent to Sb(t) = e
− t (∂x−b)
in L2(R) when t > 0. One would be
inclined to believe that this was a special property of the KdV equation. However, his is not the case.
The effect is due to the dispersive nature of the linear part of the equation. Kruzkov and Faminskii [21]
proved that u(x, 0) = u0(x) ∈ L
2(R) such that xα u0(x) ∈ L
2((0, +∞)), the weak solution of the KdV
equation, has l-continuous space derivatives for all t > 0 if l < 2α. The proof of this result is based on
the asymptotic behavior of the Airy function and its derivatives, and on the smoothing effect of the KdV
equation which was found in [17, 21]. While the proof of Kato appears to depend on special a priori
estimates, some of this mystery has been solved by the result of local gain of finite regularity for various
others linear and nonlinear dispersive equations due to Ginibre and Velo [12] and others. However, all of
them require growth conditions on the nonlinear term.
In 1992, W. Craig, T. Kappeler and W. Strauss [8, 9] proved for the fully nonlinear KdV equation
ut+ f(uxxx, uxx, ux, u, x, t) = 0, x ∈ R, t > 0 and certain additional assumption over f that C
solutions u(x, t) are obtained for all t > 0 if the initial data u0(x) decays faster than polynomially on
R+ = {x ∈ R : x > 0} and has certain initial Sobolev regularity. Following this idea, H. Cai [4] stud-
ied the nonlinear equation of KdV-type of the form ut + uxxx + a(x, , t) f(uxx, ux, u, x, t) = 0, where
a(x, t) is positive and bounded, obtaining the same conclusion. Subsequent works were given by O. Vera
[30, 31, 32, 33] for a nonlinear dispersive evolution equation, a KdV-Burgers type equation and for KdV-
Kawahara type equation, respectively. In more than one spatial dimension, J. Levandosky [23], proved
infinite gain in regularity results for nonlinear third-order equations. While [8] included local smoothing
results for some mth-order dispersive equation in n spatial dimension, their results and the techniques are
different from those presented by Levandosky. First, they consider equations with only a mild solution
and Levandosky considers equations with very general nonlinearities including a fully nonlinear equation
of the form
ut + f(D
3u, D2u, Du, u, x, t) = 0,
u(x, y, 0) = u0(x, y).
Secondly, they indicate local gain in finite regularity and Levandosky proved complementary results
showing the relationship between the decay at infinity of the initial data and the amount of gain in
regularity. More specifically, it is proved a condition under which an equation of the form
ut + a uxxx + b uxxy + c uxyy + d uyyy + f(D
2u, Du, u, x, t) = 0,
u(x, y, 0) = u0(x, y),
where a, b, c, d are assumed constant. Indeed, Levandosky proved sufficient conditions on this equation
for which a solution u will experience an infinite gain in regularity. Specifically, prove conditions for
which initial data u0(x, y) possessing sufficient decay at infinity and a minimal amount of regularity will
lead to a unique solution u(t) ∈ C∞(R2) for T ∗ where T ∗ is the existence time of solutions. According
to the characteristics of equations (P ) and considering the particular cases (Q1) and (Q2) we could hope
that the (P ) equation have gain in regularity following the steps of N. Hayashi et al. [13] or W. Craig et
al. [8].
In our problem, the initial idea is to apply the technique given by N. Hayashi et al. [13, 14] to obtain gain
in regularity. Firstly, using straightforward calculus we can see that the equation (P ) has conservation of
the energy, i. e., ||u||L2(R) = ||u0||L2(R). On the other hand, we look for estimates for ux that will help to
obtain a priori estimates, basically to obtain estimates in L∞(R). Indeed, differentiating in the x-variable
the equation (P ) we have
i ux t + i β uxxxx + ω uxxx + (|u|
2)x u+ |u|
2 ux = 0, (1.1)
and multiplying (1.1) by ux
i ux ux t + i β ux uxxxx + ω ux uxxx + (|u|
2)x u ux + |u|
2 |ux|
2 = 0
− i ux ux t − i β ux uxxxx + ω ux uxxx + (|u|
2)x uux + |u|
2 |ux|
2 = 0. (applying conjugate)
Subtracting and integrating over x ∈ R, we have
2dx+ i β
ux uxxxxdx+ i β
ux uxxxxdx
+ 2 i ω Im
ux uxxxdx+ 2 i Im
(|u|2)x u uxdx = 0.
Performing integration by parts and straightforward calculations we obtain
2dx+ 2 Im
(|u|2)x u uxdx = 0 (E1)
where
||ux||
L2(R) + 2 Im
u2 u2xdx = 0 (E2)
or integrating by parts the second term in (E1) we obtain
||ux||
L2(R) − 2 Im
|u|2 u uxxdx = 0. (E3)
Thus it is not possible to estimate in H1(R), because it appears a second term with two derivatives. The
reason of having an estimate in the derivative is related to Sobolev embedding. In one spatial dimension
we have the embedding H1(R) →֒ L∞(R). It seems that the term i β uxxx is crucial. It makes the two
”top” terms look like KdV equation; that is, ut + uxxx + . . . . Of course, the solution is complex, so that
the equation is like two coupled real KdV equations.
This was our motivation to obtain gain in regularity using the idea of W. Craig et al. [8]. We prove
conditions on (P ) for which initial data u0(x) possessing sufficient decay at infinity and a minimal amount
of regularity will lead to a unique solution u(t) ∈ C∞(R) for t > 0. We use a technique of nonlinear
multipliers, generalizing Kato’s original method, together with ideas of Craig and Goodman [7] All the
physically significant dispersive equations and systems known to us have linear parts displaying this local
smoothing property. To mention only a few, the KdV, Benjamin-Ono, intermediate long wave, various
Boussinesq, and Schrödinger equation are included. This paper is organized as follows: Section 2 outlines
briefly the notation and terminology to be used subsequently. In section 3 we prove the main inequality.
In section 4 we prove an important a priori estimate. In section 5 we prove a basic-local-in-time existence
and uniqueness theorem. In section 6 we prove a basic global existence theorem. In section 7 we develop
a series of estimates for solutions of equations (P ) in weighted Sobolev norms. These provide a starting
point for the a priori gain of regularity. In section 8 we prove the following theorem:
Theorem 1.1(Main Theorem). Let |ω| < 3 β, T > 0 and u(x, t) be a solution of (P ) in the region
R× [0, T ] such that
u ∈ L∞([0, T ] : H3(W0 L 0)) (1.2)
for some L ≥ 2. Then
u ∈ L∞([0, T ] : H3+l(Wσ, L−l, l)) ∩ L
2([0, T ] : H4+l(Wσ, L−l−1, l)) (1.3)
for all 0 ≤ l ≤ L− 1 and all σ > 0.
Remark. We consider the Gauge transformation
u(x, t) = ei d2 x+i d3 t v (x− d1 t, t) ≡ e
θ v (η, ξ) (1.4)
where θ = i d2 x+ i d3 t, η = x− d1 t and ξ = t. Then
ut = i d3 e
θ v − d1 e
θ vη + e
θ vξ : ux = i d2 e
θ v + eθ vη
uxx = − d
θ v + 2 i d2 e
θ vη + e
θ vη η : uxxx = − i d
θ v − 3 d22 e
θ vη + 3 i d2 e
θ vηη + e
θ vηηη.
Replacing in (Q) we have
− d3 e
θ v − i d1 e
θ vη + i e
θ vξ − ω d
θ v + 2 i ω d2 e
θ vη + ω e
θ vηη
β d33 e
θ v − 3 i β d22 e
θ vη − 3 β d2 e
θ vηη + i β e
θ vηηη + γ |v|
2 eθ v
− δ d2 |v|
2 eθ v + i δ |v|2 eθ vη + ǫ d2 e
θ v2v + i ǫ eθ v2 vη = 0
where
i vξ + (ω − 3 β d2) vηη + i β vηηη + (2 i ω d2 − 3 i β d
2 − i d1 + i δ |v|
2 + i ǫ v2) vη
(β d32 − ω d
2 − d3 + γ |v|
2 − δ d2 |v|
2) v + ǫ d2 v
2v = 0
: d2 =
: d3 =
− 2ω3
27 β2
. (1.5)
This way in (Q) we obtain
i vξ + i β vηηη + i (δ |v|
2 + ǫ v2) vη +
|v|2v +
v2v = 0,
but v2 v = v v v = |v|2v, then using the Gauge transformation we have the equivalent problem to (Q)
i vξ + i β vηηη + i δ |v|
2 vη + i ǫ v
2 vη +
γ + ǫ δ
− ω δ
|v|2v = 0 η, ξ ∈ R
v(η, 0) = e− i
ηu0(η).
Here, rescaling the equation, we take β = 1.
i vt + i vxxx + i δ |v|
2 vx + i ǫ v
2 vx +
γ + ǫ δ
− ω δ
|v|2v = 0 x, t ∈ R
v(x, 0) = e− i
xu0(x).
The above Gauge transformation is a bicontinuous map from Lp([0, T ] : Hs(Wσ i k)) to itself, as far as
0 < T < +∞ and p, s, σ, i, k used in this paper. With this, the assumption |ω| < 3 β imposed in Theorem
1.1 can be removed.
2 Preliminaries
We consider the initial value problem
i ut + ω uxx + i β uxxx + |u|
2 u = 0, x, t ∈ R
u(x, 0) = u0(x)
where ω, β ∈ R, β 6= 0 and u = u(x, t) is a complex valued function.
Notation. We write ∂ = ∂/∂x, ∂t = ∂/∂t and we abbreviate uj = ∂
Definition 2.1. A function ξ = ξ(x, t) belongs to the weight class Wσ i k if it is a positive C
∞ func-
tion on R× [0, T ], ∂ξ > 0 and there are constant cj , 0 ≤ j ≤ 5 such that
0 < c1 ≤ t
− k e−σ x ξ(x, t) ≤ c2 ∀ x < −1, 0 < t < T. (2.1)
0 < c3 ≤ t
− k x− i ξ(x, t) ≤ c4 ∀ x > 1, 0 < t < T. (2.2)(
t | ∂tξ | + | ∂
/ξ ≤ c5 ∀ (x, t) ∈ R× [0, T ], ∀ j ∈ N. (2.3)
Remark. We shall always take σ ≥ 0, i ≥ 1 and k ≥ 0.
Example. Let
ξ(x) =
1 + e−1/x for x > 0
1 for x ≤ 0
then ξ ∈ W0 i 0.
Definition 2.2. Let N be a positive integer. By HN (Wσ i k) we denote the Sobolev space on R with
a weight; that is, with the norm
||v||2HN (Wσ i k) =
|∂jv(x)|2 ξ(x, t) dx < +∞
for any ξ ∈ Wσ i k and 0 < t < T., Even though the norm depends on ξ, all such choices leads to equivalent
norms.
Remark. Hs(Wσ i k) depends on t (because ξ = ξ(x, t)).
Lemma 2.1. (See [4]) For ξ ∈ Wσ i 0 and σ ≥ 0, i ≥ 0, there exists a constant c > 0 such that,
for u ∈ H1(Wσ i 0),
||ξ u2|| ≤ c
|u|2 + |∂u|2
Lemma 2.2(The Gagliardo-Nirenberg inequality). Let q, r be any real numbers satisfying 1 ≤ q, r ≤ ∞
and let j and m be nonnegative integers such that j ≤ m. Then
||∂ju||Lp(R) ≤ c ||∂
mu||aLr(R) ||u||
Lq(R)
where 1
= j + a
(1−a)
for all a in the interval j
≤ a ≤ 1, and M is a positive constant
depending only on m, j, q, r and a.
Definition 2.3. By L2([0, T ] : HN (Wσ i k)) we denote the space of functions v(x, t) with the norm
(N integer positive)
||v||2L2([0, T ]:HN (Wσ i k)) =
||v(x, t)||2HN (Wσ i k)dt < +∞
Remark. The usual Sobolev space is HN (R) = HN (W0 0 0) without a weight.
Remark. We shall derive the a priori estimates assuming that the solution is C∞, bounded as x → −∞,
and rapidly decreasing as x → +∞, together with all of its derivatives.
Considering the above notation, the higher order nonlinear Schrödinger equation can be written as
i ut + i β u3 + ω u2 + |u|
2 u = 0, x, t ∈ R (2.4)
where ω, β ∈ R, β 6= 0 and u = u(x, t) is a complex valued function.
Throughout this paper c is a generic constant, not necessarily the same at each occasion(it will change
from line to line), which depends in an increasing way on the indicated quantities. In this part, we only
consider the case t > 0. The case t < 0 can be treated analogously.
3 Main Inequality
Lemma 3.1. Let |ω| < 3 β. Let u be a solution of (2.4) with enough Sobolev regularity (for instance,
u ∈ HN(R), N ≥ α+ 3), then
ξ |uα|
η |uα+1|
θ |uα|
Rαdx ≤ 0 (3.1)
where
η = (3 β − |ω|) ∂ξ for |ω| < 3 β
θ = − [ ∂tξ + β ∂
3ξ + |ω| ∂ξ + c0 ξ ] where c0 = ||u||
L∞(R)
and Rα = Rα(|uα|, |uα−1|, . . .).
Proof. Differentiating (2.4) α-times (for α ≥ 0) over x ∈ R leads to
i uα t + i β uα+3 + ω uα+2 + (|u|
2)α u+
(|u|2)α−m um + |u|
2 uα = 0. (3.2)
Let ξ = ξ(x, t), then multiplying (3.2) by ξ uα we have
i ξ uα uα t + i β ξ uα uα+3 + ω ξ uα uα+2 + (|u|
2)α ξ u uα
(|u|2)α−m ξ um uα + ξ |u|
2 |uα|
2 = 0
− i ξ uα uα t − i β ξ uα uα+3 + ω ξ uα uα+2 + (|u|
2)α ξ u uα
(|u|2)α−m ξ um uα + ξ |u|
2 |uα|
2 = 0. (applying conjugate)
Subtracting and integrating over x ∈ R we have
ξ |uα|
2dx+ i β
ξ uα uα+3dx+ i β
ξ uα uα+3dx− i
ξt |uα|
ξ uα uα+2dx − ω
ξ uα uα+2dx+ 2 i Im
ξ (|u|2)α u uαdx
+ 2 i
ξ (|u|2)α−m um uαdx = 0. (3.3)
We estimate the second term integrating by parts
ξ uα uα+3dx =
∂2ξ uα uα+1dx+ 2
∂ξ |uα+1|
ξ uα+2 uα+1dx.
The other terms are calculated in a similar way. Hence, replacing in (3.3) and performing straightforward
calculations we obtain
ξ |uα|
2dx+ i β
∂2ξ uα uα+1dx+ 2 i β
∂ξ |uα+1|
+ i β
ξ uα+2 uα+1dx+ i β
∂2ξ uα uα+1dx+ i β
∂ξ |uα+1|
− i β
ξ uα+1 uα+2dx− ω
∂ξ uα uα+1dx− ω
ξ |uα+1|
∂ξ uα uα+1dx+ ω
ξ |uα+1|
2dx− i
∂tξ |uα|
+ 2 i Im
ξ (|u|2)α u uαdx+ 2 i
ξ (|u|2)α−m um uαdx = 0
ξ |uα|
2dx− β
∂3ξ |uα|
2dx+ 3 β
∂ξ |uα+1|
2dx − 2ω Im
∂ξ uα uα+1dx
∂tξ |uα|
2dx+ 2 Im
ξ (|u|2)α u uαdx+ 2
ξ (|u|2)α−m um uαdx = 0
hence
ξ |uα|
2dx− β
∂3ξ |uα|
2dx+ 3 β
∂ξ |uα+1|
2dx+ 2 Im
(|u|2)α ξ u uαdx
∂tξ |uα|
2dx+ 2
ξ (|u|2)α−m um uαdx = 2ω Im
∂ξ uα uα+1dx
≤ |ω|
∂ξ |uα|
2dx+ |ω|
∂ξ |uα+1|
therefore
ξ |uα|
[ 3 β − |ω| ] ∂ξ |uα+1|
[ ∂tξ + β ∂
3ξ + |ω| ∂ξ ] |uα|
+ 2 Im
(|u|2)α ξ u uαdx+ 2
ξ (|u|2)α−m um uαdx ≤ 0. (3.4)
(|u|2)α = (u u )α =
uα−k uk = uuα +
uα−k uk + u uα
(|u|2)α u uα = |u|
2|uα|
uα−k uk u uα + u
2 u2α
thus,
( |u|2)α ξ u uαdx = 2
ξ uα−k uk u uαdx+ 2 Im
ξ u2 u2αdx
ξ |uα−k| |uk| |u| |uα|dx+ 2
ξ |u|2 |uα|
ξ |uα−k| |uk| |u| |uα|dx+ 2 ||u||
L∞(R)
ξ |uα|
≤ 2 ||u||L∞(R)
ξ |uα−k| |uk| |uα|dx+ 2 ||u||
L∞(R)
ξ |uα|
2dx (3.5)
hence, in (3.4) we have
ξ |uα|
[3 β − |ω| ] ∂ξ |uα+1|
[∂tξ + β ∂
3ξ + |ω| ∂ξ + c0 ξ ] |uα|
− 2 c
ξ |uα−k| |uk| |uα|dx− 2
ξ |(|u|2)α−m| |um| |uα|dx ≤ 0.
Therefore, using straightforward calculations we obtain the main inequality
ξ |uα|
η |uα+1|
θ |uα|
Rαdx ≤ 0
where
η = (3 β − |ω| ) ∂ξ for |ω| < 3 β
θ = − [ ∂tξ + β ∂
3ξ + |ω| ∂ξ + c0 ξ ] where c0 = ||u||
L∞(R)
and Rα = Rα(|uα|, |uα−1|, . . .).
Remark. In (3.4) using Young’s estimate and assuming that β > 0 we have
2ω Im
uα uα+1 dx ≤
2 dx+ 2 β
|uα+1|
2 dx.
Then, in (3.4) we obtain
ξ |uα|
2dx− β
∂3ξ |uα|
2dx+ β
∂ξ |uα+1|
2dx + 2 Im
(|u|2)α ξ u uαdx
∂tξ |uα|
2dx+ 2
ξ (|u|2)α−m um uαdx = 2ω Im
∂ξ uα uα+1dx
and the assumption that |ω| < 3 β can be removed.
Lemma 3.2. For η ∈ Wσ i k an arbitrary weight function and |ω| < 3 β, there exists ξ ∈ Wσ, i+1, k
that satisfies
η = (3 β − |ω|) ∂ξ for |ω| < 3 β. (3.6)
Indeed, we have
(3 β − |ω|)
η(y, t) dy. (3.7)
Lemma 3.3. The expression Rα in the inequality of Lemma 3.1 is a sum of terms of the form
ξ uν1 uν2 uα (3.8)
where 1 ≤ ν1 ≤ ν2 ≤ α and
ν1 + ν2 = α (3.9)
Proof. It follows from (3.5).
4 An a priori estimate
We show now a fundamental a priori estimate used for a basic local-in-time existence theorem. We con-
struct a mapping Z : L∞([0, T ] : Hs(R)) 7−→ L∞([0, T ] : Hs(R)) with the property:
Given u(n) = Z(u(n−1)) and essupt∈[0, T ]||u
(n−1)||s ≤ c0 then essupt∈[0, T ]||u
(n)||s ≤ c0, where s and
c0 > 0 are constants. This property tells us that Z : Bc0(0) 7−→ Bc0(0) where Bc0(0) = {v(x, t) :
||v( · , t)||s ≤ c0} is a ball in L
∞([0, T ] : Hs(R)). To guarantee this property, we will appeal to an a priori
estimate which is the main object of this section.
Differentiating (2.4) two times leads to
i ∂tu2 + i β u5 + ω u4 + (|u|
2)2 u+ 2 (|u|
2)1 u1 + |u|
2 u2 = 0. (4.1)
Let u = ∧v where ∧ = (I − ∂2)−1. Hence u = (I − ∂2)−1v then u− u2 = v where ∂tu2 = − vt + ut.
Replacing in (4.1) we have
− i vt + i β ∧ v5 + ω ∧ v4 + (| ∧ v|
2)2 ∧ v + 2 (| ∧ v|
2)1 ∧ v1
+ | ∧ v|2 ∧ v2 − (i β ∧ v3 + ω ∧ v2 + | ∧ v|
2 ∧ v) = 0. (4.2)
The (4.2) equation is linearized by substituting a new variable z in each coefficient:
− i vt + i β ∧ v5 + ω ∧ v4 + (| ∧ z|
2)2 ∧ v + 2 (| ∧ z|
2)1 ∧ v1
+ | ∧ z|2 ∧ v2 − (i β ∧ v3 + ω ∧ v2 + | ∧ z|
2 ∧ v) = 0. (4.3)
The linear equation which is to be solved at each iteration is of the form
i ∂tv = i β ∧ v
5 + ω ∧ v
4 − i β ∧ v
3 − ω ∧ v
2 + b
(1) (4.4)
where b(1) = (| ∧ z|2)2 ∧ v+2 (| ∧ z|
2)1 ∧ v1 + | ∧ z|
2 ∧ v2 − |∧ z|
2 ∧ v. Equation (4.4) is a linear equation
at each iteration which can be solved in any interval of time in which the coefficient is defined.
We consider the following lemma that will help us setting up the iteration scheme.
Lemma 4.1. Let |ω| < 3 β. Given initial data u0(x) ∈ H
∞(R) =
N≥0 H
N (R) there exists a unique
solution of (4.4) where b(1) is a smooth bounded coefficient with z ∈ H∞(R). The solution is defined in
any time interval in which the coefficient is defined.
Proof. Let T > 0 be arbitrary and M > 0 a constant. Let
Γ = ξ ( i ∂t − i β ∧ ∂
5 − ω ∧ ∂4 + i β ∧ ∂3 + ω ∧ ∂2 )
then in (4.4) we have Γu = ξ b(1). We consider the bilinear form B : D ×D 7−→ R,
B(u, v) =< u, v >= Im
e−Mt u v dx dt
where D = {u ∈ C∞0 (R× [0, T ]) : u(x, 0) = 0 }. We have
Γu · u = i ξ u ut − i β ξ u ∧ u5 − ω ξ u ∧ u4 + i β ξ u ∧ u3 + ω ξ u ∧ u2
Γu · u = − i ξ u ut + i β ξ u ∧ u5 − ω ξ u ∧ u4 − i β ξ u ∧ u3 + ω ξ u ∧ u2. (applying conjugate)
Subtracting and integrating over x ∈ R we have
2 i Im
Γu · udx = i ∂t
ξ |u|2dx− i
∂tξ |u|
2dx− i β
ξ u ∧ u5dx− i β
ξ u ∧ u5dx
ξ u ∧ u4dx+ ω
ξ u ∧ u4dx+ i β
ξ u ∧ u3dx+ i β
ξ u ∧ u3dx
ξ u ∧ u2dx− ω
ξ u ∧ u2dx.
Each term is treated separately, integrating by parts
ξ u ∧ u5dx =
ξ ∧ (I − ∂2)u ∧ u5dx =
ξ ∧ u ∧ u5dx−
ξ ∧ u2 ∧ u5dx
∂4ξ ∧ u ∧ u1dx+
∂3ξ | ∧ u1|
2dx− 3
∂2ξ ∧ u1 ∧ u2dx− 2
∂ξ | ∧ u2|
ξ ∧ u2 ∧ u3dx−
∂2ξ ∧ u2 ∧ u3dx−
∂ξ | ∧ u3|
ξ ∧ u3 ∧ u4dx.
The other terms are calculates in a similar way. Then
2 i Im
Γu · udx
= i ∂t
ξ |u|2dx− i
∂tξ |u|
2dx− i β
∂4ξ ∧ u ∧ u1dx− i β
∂3ξ | ∧ u1|
+ 3 i β
∂2ξ ∧ u1 ∧ u2dx+ 2 i β
∂ξ | ∧ u2|
2dx− i β
ξ ∧ u2 ∧ u3dx
+ i β
∂2ξ ∧ u2 ∧ u3dx+ i β
∂ξ | ∧ u3|
2dx− i β
ξ ∧ u3 ∧ u4dx
− i β
∂4ξ ∧ u ∧ u1dx − i β
∂3ξ | ∧ u1|
2dx+ 3 i β
∂2ξ ∧ u1 ∧ u2dx
+ 2 i β
∂ξ | ∧ u2|
2dx− i β
ξ ∧ u2 ∧ u3dx+ i β
∂2ξ ∧ u2 ∧ u3dx
+ 2 i β
∂ξ | ∧ u3|
2dx+ i β
ξ ∧ u3 ∧ u4dx+ ω
∂3ξ ∧ u ∧ u1dx
∂2ξ | ∧ u1|
2dx− 2ω
∂ξ ∧ u1 ∧ u2dx− ω
ξ | ∧ u2|
∂ξ ∧ u2 ∧ u3dx− ω
ξ | ∧ u3|
2dx − ω
∂3ξ ∧ u ∧ u1dx
∂2ξ | ∧ u1|
2dx+ 2ω
∂ξ ∧ u1 ∧ u2dx+ ω
ξ | ∧ u2|
∂ξ ∧ u2 ∧ u3dx+ ω
ξ | ∧ u3|
2dx + i β
∂2ξ ∧ u ∧ u1dx
+ i β
∂ξ | ∧ u1|
2dx− i β
ξ ∧ u1 ∧ u2dx− i β
ξ ∧ u2 ∧ u3dx
+ i β
∂2ξ ∧ u ∧ u1dx + i β
∂ξ | ∧ u1|
2dx− i β
ξ ∧ u1 ∧ u2dx
− i β
ξ ∧ u2 ∧ u3dx− ω
∂ξ ∧ u ∧ u1dx− ω
ξ | ∧ u1|
2dx− ω
ξ | ∧ u2|
∂ξ ∧ u ∧ u1dx+ ω
ξ | ∧ u1|
2dx+ ω
ξ | ∧ u2|
hence
2 i Im
Γu · udx = i ∂t
ξ |u|2dx− i
∂tξ |u|
2dx− i β
∂4ξ (| ∧ u|2)1dx
− 2 i β
∂3ξ | ∧ u1|
2dx+ 3 i β
∂2ξ (| ∧ u1|
2)1dx + 4 i β
∂ξ | ∧ u2|
− i β
ξ (| ∧ u2|
2)1dx+ i β
∂2ξ (| ∧ u2|
2)1dx+ 3 i β
∂ξ | ∧ u3|
+2 i ω Im
∂3ξ ∧ u ∧ u1dx − 4 i ω Im
∂ξ ∧ u1 ∧ u2dx
− 2 i ω Im
∂ξ ∧ u2 ∧ u3dx+ i β
∂2ξ (| ∧ u|2)1dx+ 2 i β
∂ξ | ∧ u1|
− i β
ξ (| ∧ u1|
2)1dx− i β
ξ (| ∧ u2|
2)1dx− 2ω Im
∂ξ ∧ u ∧ u1dx
then, adding similar terms and cutting the letter i we obtain
Γu · u dx = ∂t
ξ |u|2dx−
∂tξ |u|
2dx + β
∂5ξ | ∧ u|2dx− 5 β
∂3ξ | ∧ u1|
+ 6 β
∂ξ | ∧ u2|
2dx− β
∂3ξ | ∧ u2|
2dx+ 3 β
∂ξ | ∧ u3|
+ 2ω Im
∂3ξ ∧ u ∧ u1dx− 4ω Im
∂ξ ∧ u1 ∧ u2dx− 2ω Im
∂ξ ∧ u2 ∧ u3dx
∂3ξ | ∧ u|2dx+ 3 β
∂ξ | ∧ u1|
2dx − 2ω Im
∂ξ ∧ u ∧ u1dx
∂ξ | ∧ u3|
2dx+ |ω|
∂ξ | ∧ u2|
2dx+ 2 |ω|
∂ξ | ∧ u1|
2dx+ 2 |ω|
∂ξ | ∧ u2|
+ |ω|
∂ξ | ∧ u|2dx+ |ω|
∂ξ | ∧ u1|
2dx+ |ω|
|∂3ξ| | ∧ u|2dx
+ |ω|
|∂3ξ| | ∧ u1|
2dx +
∂tξ |u|
2dx+ 2 Im
Γu · udx
ξ |u|2dx+ 3 β
∂ξ | ∧ u3|
2dx− β
∂3ξ | ∧ u2|
2dx+ 6 β
∂ξ | ∧ u2|
− 5 β
∂3ξ | ∧ u1|
2dx+ 3 β
∂ξ | ∧ u1|
2dx+ β
∂5ξ | ∧ u|2dx − β
∂3ξ | ∧ u|2dx
where
3 |ω|
∂ξ | ∧ u2|
2dx+ |ω|
[|∂3ξ|+ 3 ∂ξ] | ∧ u1|
+ |ω|
[|∂3ξ|+ ∂ξ + ∂tξ] | ∧ u|
2dx+ 2 Im
Γu · udx
ξ |u|2dx +
[3 β − |ω|] ∂ξ | ∧ u3|
2dx− β
∂3ξ | ∧ u2|
+ 6 β
∂ξ | ∧ u2|
2dx− 5 β
∂3ξ | ∧ u1|
2dx+ 3 β
∂ξ | ∧ u1|
∂5ξ | ∧ u|2dx− β
∂3ξ | ∧ u|2dx
ξ |u|2dx + β
[−∂3ξ + 5∂ξ] | ∧ u2|
[−5 ∂3ξ + 3∂ξ] | ∧ u1|
2dx+ β
[∂3ξ − ∂3ξ] | ∧ u|2dx
using (2.3), ∧un = (I − (I − ∂
2))∧ un−2 = ∧un−2 − un−2 for n a positive integer and standard estimates
we obtain
Γu · u dx ≥ ∂t
ξ |u|2 dx− c
ξ |u|2 dx.
Multiply this equation by e−Mt, and integrate with respect to t for t ∈ [0, T ] and u ∈ D
e−Mt Γu · udx dt ≥
ξ |u|2dx
dt− c
ξ e−Mt |u|2dx dt
= e−Mt
ξ |u|2dx
ξ e−Mt |u|2dx dt− c
ξ e−Mt |u|2dx dt
= e−Mt
ξ(x, T ) |u(x, T )|2dx+M
ξ e−Mt |u|2dx dt − c
ξ e−Mt |u|2dx dt.
< Γu, u >= Im
e−Mt Γu · udx dt
≥ e−Mt
ξ(x, T ) |u(x, T )|2dx+ (M − c)
ξ e−Mt |u|2dx dt
ξ e−Mt |u|2dx dt
provided that M is chosen large enough. Then < Γu, u >≥< u, u >, for all u ∈ D. Let Γ∗ be the
formal adjoint of Γ defined by Γ∗ = ξ(−i ∂t − i β ∧ ∂
5 − ω ∧ ∂4 + i β ∧ ∂3 + ω ∧ ∂2). Let D∗ = {w ∈
C∞0 (R× [0, T ]) : w(x, T ) = 0 }. In a similar way we prove that
< Γ∗w, w > ≥ < w, w >, ∀ w ∈ D∗.
¿From this equation, we have that Γ∗ is one-one. Therefore, < Γ∗w, Γ∗v > is an inner product on D∗. We
denote by X the completion of D∗ with respect to this inner product. By Riesz’s Representation Theo-
rem, there exists a unique solution V ∈ X, such that for any w ∈ D∗, < ξb(1), w >=< Γ∗V, Γ∗w > where
we use that ξ b(1) ∈ X. Then if v = Γ∗V we have < v, Γ∗w >=< ξb(1), w > or < Γ∗w, v >=< w, ξb(1) > .
Hence, v = Γ∗V is a weak solution of Γv = ξb(1) with v ∈ L2(R× [0, T ]) ≃ L2([0, T ] : L2(R)).
Remark. To obtain higher regularity of the solution, we repeat the proof with higher derivatives. It
is a standard approximation procedure to obtain a result for general initial data.
The next step is to estimate the corresponding solutions v = v(x, t) of the equation (4.3) via the coeffi-
cients of that equation.
The following estimate is related to the existence of solutions theorem.
Lemma 4.2. Let |ω| < 3 β and 0 < γ1 ≤ ξ ≤ γ2, with γ2, γ2 real constants. Let v, z ∈ C
k([0, +∞) :
HN(R)) for all k, N which satisfy (4.3). For each integer α there exist positive nondecreasing functions
G and F such that for all t ≥ 0
ξ |vα|
2dx ≤ G(||z||λ) ||v||
α + F (||z||α) (4.5)
where || · ||α is the norm in H
α(R) and λ = max{1, α}.
Proof. Differentiating α-times the equation (4.3), for some α ≥ 0 we have
− i ∂tvα + i β ∧ vα+5 + ω ∧ vα+4 − i β ∧ vα+3 +
h(j) ∧ vj + (|z|
2)α+2 ∧ v + p(∧zα+1, . . .) = 0 (4.6)
where h(j) is a smooth function depending on | ∧ z|2, . . . with i = 2 + α − j. For α ≥ 2, p(∧zα+1, . . .)
depends at most linearly on ∧zα+1, while for α = 2, p(∧zα+1, . . .) depends at most quadratically on
∧zα+1.
We multiply equation (4.6) by ξ vα and integrate over x ∈ R
ξ vα ∂tvαdx+ i β
ξ vα ∧ vα+5dx+ ω
ξ vα ∧ vα+4dx− i β
ξ vα ∧ vα+3dx
ξ vα ∧ vjdx+
ξ (|z|2)α+2vα ∧ vdx+
ξ vαp(∧zα+1, . . .)dx = 0
and applying conjugate
ξ vα ∂tvαdx− i β
ξ vα ∧ vα+5dx+ ω
ξ vα ∧ vα+4dx+ i β
ξ vα ∧ vα+3dx
ξ vα ∧ vjdx+
ξ (|z|2)α+2vα ∧ vdx+
ξ vαp(∧zα+1, . . .)dx = 0.
Subtracting, it follows that
− i ∂t
ξ |vα|
2dx+ i
∂tξ |vα|
2dx+ i β
ξ vα ∧ vα+5dx+ i β
ξ vα ∧ vα+5dx
ξ vα ∧ vα+4dx− ω
ξ vα ∧ vα+4dx− i β
ξ vα ∧ vα+3dx− i β
ξ vα ∧ vα+3dx
ξ vα ∧ vjdx−
ξ vα ∧ vjdx+
ξ (|z|2)α+2vα ∧ vdx (4.7)
ξ (|z|2)α+2vα ∧ v dx+
ξ vαp(∧zα+1, . . .) dx−
ξ vα p(∧zα+1, . . .)dx = 0.
Each term is treated separately, integrating by parts
ξ vα ∧ vα+5dx =
ξ ∧ (I − ∂2)vα ∧ vα+5dx
ξ ∧ vα ∧ vα+5dx−
ξ ∧ vα+2 ∧ vα+5dx
∂4ξ ∧ vα ∧ vα+1dx+
∂3ξ | ∧ vα+1|
2dx− 3
∂2ξ ∧ vα+1 ∧ vα+2dx
∂ξ | ∧ vα+2|
ξ ∧ vα+2 ∧ vα+3dx−
∂2ξ ∧ vα+2 ∧ vα+3dx
∂ξ | ∧ vα+3|
ξ ∧ vα+4 ∧ vα+3dx.
The other terms are calculated in a similar way. Hence in (4.7) we have performing straightforward
calculations as above
ξ |vα|
∂tξ |vα|
2dx − β
∂5ξ | ∧ vα|
2dx+ 2 β
∂3ξ | ∧ vα+1|
∂3ξ | ∧ vα+1|
2dx− 4 β
∂ξ | ∧ vα+2|
2dx− β
∂ξ | ∧ vα+2|
∂2ξ | ∧ vα+2|
2dx− 3 β
∂ξ | ∧ vα+3|
2 dx− 2ω Im
∂3ξ ∧ vα ∧ vα+1dx
+ 4ω Im
∂ξ ∧ vα+1 ∧ vα+2dx+ 2ω Im
∂ξ ∧ vα+2 ∧ vα+3dx
+ 2 β Im
∂ξ ∧ vα ∧ vα+2dx+ 2 β Im
ξ ∧ vα+1 ∧ vα+2dx
∂ξ | ∧ vα+2|
2dx+ 2
h(j) Im
ξ vα ∧ vjdx
+ 2 Im
ξ (|z|2)α+2 vα ∧ vdx+ 2 Im
ξ vαp(∧zα+1, . . .) dx = 0
ξ |vα|
∂tξ |vα|
2dx − 3 β
∂ξ | ∧ vα+3|
2dx+ β
∂2ξ | ∧ vα+2|
− 6 β
∂ξ | ∧ vα+2|
2dx+ 5 β
∂3ξ | ∧ vα+1|
2dx− β
∂5ξ | ∧ vα|
= − 2ω Im
∂ξ ∧ vα+2 ∧ vα+3dx− 4ω Im
∂ξ ∧ vα+1 ∧ vα+2dx
− 2 β Im
ξ ∧ vα+1 ∧ vα+2dx− 2 β Im
∂ξ ∧ vα ∧ vα+2dx
+2ω Im
∂3ξ ∧ vα ∧ vα+1dx− 2
h(j) Im
ξ vα ∧ vjdx
− 2 Im
ξ (|z|2)α+2vα ∧ vdx− 2 Im
ξ vαp(∧zα+1, . . .) dx
hence,
ξ |vα|
∂tξ |vα|
2dx+ 3 β
∂ξ | ∧ vα+3|
2dx− β
∂2ξ | ∧ vα+2|
∂ξ | ∧ vα+2|
2dx− 5 β
∂3ξ | ∧ vα+1|
2dx+ β
∂5ξ | ∧ vα|
= 2ω Im
∂ξ ∧ vα+2 ∧ vα+3dx+ 4ω Im
∂ξ ∧ vα+1 ∧ vα+2dx
+2 β Im
ξ ∧ vα+1 ∧ vα+2dx + 2 β Im
∂ξ ∧ vα ∧ vα+2dx
− 2ω Im
∂3ξ ∧ vα ∧ vα+1dx+ 2
h(j) Im
ξ vα ∧ vjdx
+ 2 Im
ξ (|z|2)α+2vα ∧ vdx+ 2 Im
ξ vαp(∧zα+1, . . .) dx
≤ |ω|
∂ξ | ∧ vα+2|
2 dx+ |ω|
∂ξ| ∧ vα+3|
2dx+ 2 |ω|
∂ξ | ∧ vα+1|
+ 2 |ω|
∂ξ| ∧ vα+2|
2dx+ |β|
ξ | ∧ vα+1|
2dx+ |β|
ξ | ∧ vα+2|
+ |β|
∂ξ| ∧ vα|
2dx+ |β|
∂ξ| ∧ vα+2|
2dx+ |ω|
∂3ξ| ∧ vα|
+ |ω|
∂3ξ| ∧ vα+1|
2dx + 2
∣∣∣∣∣∣
ξ vα ∧ vjdx
∣∣∣∣∣∣
ξ (|z|2)α+2 vα ∧ vdx
ξ vαp(∧zα+1, . . .) dx
where
ξ |vα|
(3 β − |ω|)∂ξ | ∧ vα+3|
2dx +
[β ∂2ξ − 6 β ∂ξ + 3 |ω| ∂ξ + |β| ∂ξ + |β| ξ] | ∧ vα+2|
[5β∂3ξ + |ω|∂3ξ + 2 |ω|∂ξ + |β| ξ] | ∧ vα+1|
[∂tξ + β ∂
5ξ + |ω| ∂3ξ + |β| ∂ξ] | ∧ vα|
∣∣∣∣∣∣
ξ vα ∧ vjdx
∣∣∣∣∣∣
ξ (|z|2)α+2 vα ∧ vdx
∣∣∣∣+ 2
ξ vαp(∧zα+1, . . .) dx
∣∣∣∣ .
using that |ω| < 3 β we have that the first term in the right hand side of the above expression is not
positive. Hence,
ξ |vα|
[β ∂2ξ − 6 β ∂ξ + 3 |ω| ∂ξ + |β| ∂ξ + |β| ξ] | ∧ vα+2|
[5 β ∂3ξ + |ω| ∂3ξ + 2 |ω| ∂ξ + |β| ξ] | ∧ vα+1|
2dx +
[∂tξ + β ∂
5ξ + |ω| ∂3ξ + |β| ∂ξ] | ∧ vα|
∣∣∣∣∣∣
ξ vα ∧ vjdx
∣∣∣∣∣∣
ξ (|z|2)α+2 vα ∧ v dx
∣∣∣∣+ 2
ξ vαp(∧zα+1, . . .) dx
∣∣∣∣ .
Using that ∧vn = ∧vn−2 − vn−2 and a standard estimate, the lemma follows.
5 Uniqueness and Existence of a Local Solution
In this section, we study the uniqueness and the existence of local strong solutions in the Sobolev space
HN(R) for N ≥ 3 for the problem (2.4). To establish the existence of strong solutions for (2.4) we use
the a priori estimate together with an approximation procedure.
Theorem 5.1(Uniqueness). Let |ω| < 3 β, u0(x) ∈ H
N (R) with N ≥ 3 and 0 < T < +∞. Then
there is at most one strong solution u ∈ L∞([0, T ] : HN(R)) of (2.4) with initial data u(x, 0) = u0(x).
Proof. Assume that u, v ∈ L∞([0, T ] : HN(R)) are two solutions of (2.4) with ut, vt ∈ L
∞([0, T ] :
HN−3(R)), and with the same initial data. Then
i (u− v)t + i β (u− v)3 + ω (u− v)2 + |u|
2 u− |v|2 v = 0 (5.1)
with (u− v)(x, 0) = 0. By (5.1)
i (u− v)t + i β (u − v)3 + ω (u − v)2 + |u|
2 (u− v) + (|u|2 − |v|2) v = 0
i (u− v)t + i β (u− v)3 + ω (u− v)2 + |u|
2 (u− v) + (|u| − |v|) (|u|+ |v|) v = 0. (5.2)
Multiplying (5.2) by ξ(u − v) we have
i ξ (u− v) (u − v)t + i β ξ (u − v) (u− v)3 + α ξ (u− v) (u − v)2
+ |u|2 |u− v|2 + ξ (u− v) (|u| − |v|) (|u|+ |v|) v = 0.
− i ξ (u− v) (u − v)t − i β ξ (u− v) (u − v)3 + α ξ (u − v) (u− v)2
+ |u|2 |u− v|2 + ξ (u− v) (|u| − |v|) (|u|+ |v|) v = 0. (applying conjugate)
Subtracting and integrating over x ∈ R we obtain
ξ |u− v|2dx− i
∂tξ |u− v|
2dx+ i β
ξ (u− v) (u− v)3dx
+ i β
ξ (u − v) (u− v)3dx+ ω
ξ (u − v) (u− v)2dx
ξ (u− v) (u − v)2dx+ 2 i Im
ξ (u − v) (|u| − |v|) (|u|+ |v|) v dx = 0 (5.3)
Each term is treated separately, integrating by parts
ξ (u− v) (u− v)3dx
∂2ξ (u− v) (u− v)1dx+ 2
∂ξ |(u − v)1|
ξ (u− v)1 (u− v)2dx.
The other terms are calculated in a similar way. Hence in (5.3) we have
ξ |u− v|2dx− i
∂tξ |u− v|
2dx+ i β
∂2ξ (u− v) (u − v)1dx
+2 i β
∂ξ |(u − v)1|
2dx+ i β
ξ (u − v)1 (u − v)2dx + i β
∂2ξ (u− v) (u − v)1dx
+ i β
∂ξ |(u− v)1|
2dx− i β
ξ (u− v)1 (u− v)2dx− ω
∂ξ (u − v) (u− v)1dx
ξ |(u− v)1|
2dx+ ω
∂ξ (u− v) (u − v)1dx+ ω
ξ |(u − v)1|
+2 i Im
ξ (u− v) (|u| − |v|) (|u|+ |v|) v dx = 0
ξ |u− v|2dx− i
∂tξ |u− v|
2dx + i β
∂2ξ (|u − v|2)1dx+ 3 i β
∂ξ |(u − v)1|
− 2 i ω Im
∂ξ (u− v) (u− v)1dx+ 2 i Im
ξ (u− v) (|u| − |v|) (|u|+ |v|) v dx = 0
if and only if
ξ |u− v|2dx−
∂tξ |u− v|
2 dx+ β
∂2ξ (|u− v|2)1dx + 3 β
∂ξ |(u− v)1|
= 2ω Im
∂ξ (u− v) (u− v)1dx− 2 Im
ξ (u− v) (|u| − |v|) (|u|+ |v|) v dx
≤ |ω|
∂ξ |u− v|2dx+ |ω|
∂ξ |(u− v)1|
2dx+ 2
ξ |u− v| | |u| − |v| | (|u|+ |v|) |v| dx.
Using that | |u| − |v| | ≤ |u− v|, (2.3) and standard estimates, we have
ξ |u− v|2dx+
[3 β − |ω| ] ∂ξ |(u− v)1|
2dx ≤ c
ξ |u− v|2dx.
Integrating in t ∈ [0, T ], using the fact that (u− v) vanishes at t = 0 and Gronwall’s inequality it follows
that u = v. This proves the uniqueness of the solution.
We construct the mapping Z : L∞([0, T ] : Hs(R)) 7−→ L∞([0, T ] : Hs(R)) where the initial condi-
tion is given by u(n)(x, 0) = u0(x) and the first approximation is given by
u(0) = u0(x)
u(n) = Z(u(n−1)) n ≥ 1,
where u(n−1) is in place of z in equation (4.3) and u(n) is in place of v which is the solution of equation
(4.3). That is
− i u
t + i β ∧ u
5 + ω ∧ u
4 + (| ∧ u
(n−1)|2)2 ∧ u
(n) + 2 (| ∧ u(n−1)|2)1 ∧ u
+ | ∧ u(n−1)|2 ∧ u
2 − (i β ∧ u
3 + ω ∧ u
2 + | ∧ u
(n−1)|2 ∧ u(n)) = 0.
By Lemma 4.1, u(n) exists and is unique in C((0, +∞) : HN (R)). A choice of c0 and the use of the
a priori estimate in Section 4 shows that Z : Bc0(0) 7−→ Bc0(0) where Bc0(0) is a bounded ball in
L∞([0, T ] : Hs(R)).
Theorem 5.2(Local solution). Let |ω| < 3 β and N an integer ≥ 3. If u0(x) ∈ H
N (R), then there
is T > 0 and u such that u is a strong solution of (2.4), u ∈ L∞([0, T ] : HN (R)) and u(x, 0) = u0(x).
Proof. We prove that for u0(x) ∈ H
∞(R) =
k≥0 H
k(R) there exists a solution u ∈ L∞([0, T ] : HN(R))
with initial data u(x, 0) = u0(x) where the time of existence T > 0 only depends on the norm of u0(x).
We define a sequence of approximations to equation (4.3) as
t = i β ∧ v
5 + ω ∧ v
4 − i β ∧ v
3 − ω ∧ v
2 + | ∧ v
(n−1)|2 ∧ v
+ O[ (| ∧ v(n−1)|2)2, (| ∧ v
(n−1)|2)1, . . .) ] (5.4)
where the initial condition is v(n)(x, 0) = u0(x)−∂
2u0(x). The first approximation is given by v
(0)(x, 0) =
u0(x)− ∂
2u0(x). Equation (5.4) is a linear equation at each iteration which can be solved in any interval
of time in which the coefficients are defined. This is shown in Lemma 4.1. By Lemma 4.2, it follows that
ξ |v(n)α |
2dx ≤ G(||v(n−1)||λ) ||v
(n)||2α + F (||v
(n−1)||α). (5.5)
Choose α = 1 and let c ≥ ||u0−∂
2u0||1 ≥ ||u0||3. For each iterate n, ||v
(n)( · , t)|| is continuous in t ∈ [0, T ]
and ||v(n)( · , 0)|| < c. Define c0 =
c2+1. Let T
0 be the maximum time such that ||v
(k)( · , t)||1 ≤ c3
for 0 ≤ t ≤ T
0 , 0 ≤ k ≤ n. Integrating (5.5) over [0, t] we have that for 0 ≤ t ≤ T
0 and j = 0, 1
||v(n−1)||1
||v(n)||2jds+
||v(n−1)||j
It follows that
ξ(x, t)|v
j (x, t)|
2dx ≤
ξ(x, 0)|v
j (x, 0)|
||v(n−1)||1
||v(n)||2jds
||v(n−1)||j
hence
j (x, t)|
2dx ≤
ξ(x, t)|v
j (x, t)|
ξ(x, 0)|v
j (x, 0)|
||v(n−1)||1
||v(n)||2jds
||v(n−1)||j
2dx ≤
j (x, 0)|
G(c3)
c23 t+
F (c3)
and we obtain for j = 0, 1 that
||v(n)||1 ≤
G(c0)
c20 t+
F (c0)
Claim. T
0 does not approach to 0.
On the contrary, assume that T
0 → 0. Since ||v
(n)( · , t)|| is continuous for t ≥ 0, there exists τ ∈ [0, T ]
such that ||v(k)( · , t)||1 = c0 for 0 ≤ τ ≤ T
0 , 0 ≤ k ≤ n. Then
c20 ≤
G(c0)
c20 T
F (c0)
as n → ∞, we have
c2 + 1
c2 then
4 γ21
c4 + 1 ≤ 0
which is a contradiction. Consequently T
0 6→ 0. Choosing T = T (c) sufficiently small, and T not
depending on n, one concludes that
||v(n)||1 ≤ C (5.6)
for 0 ≤ t ≤ T. This shows that T
0 ≥ T. Hence, from (5.6) we imply that there exists a subsequence
v(nj) ≡ v(n) such that
⇀ v weakly on L∞([0, T ] : H1(R)). (5.7)
Claim. u = ∧v is a solution.
In the linearized equation (5.4) we have
5 = ∧(I − (I − ∂
3 = ∧v
3 − v
3 = ∂
1︸ ︷︷ ︸
∈L2(R)
)− ∂2(v
1 )︸ ︷︷ ︸
∈H−2(R)
∈ H−2(R).
Since ∧ = (I − ∂2)−1 is bounded in H1(R), ∧v
5 belongs to H
−2(R). v(n) is still bounded in
L∞([0, T ] : H1(R)) →֒ L2([0, T ] : H1(R)) and since ∧ : L2(R) → H2(R) is a bounded operator,
|| ∧ v
1 ||H2(R) ≤ c ||v
1 ||L2(R) ≤ c ||v
1 ||H1(R).
Consequently, ∧v
1 is bounded in L
2([0, T ] : H2(R)) →֒ L2([0, T ] : L2(R)). It follows that ∂2(∧v
1 ) is
bounded in L2([0, T ] : H−2(R)), and
5 is bounded in L
2([0, T ] : H−2(R)). (5.8)
Similarly, the other terms are bounded. By (5.4), v
t is a sum of terms each of which is the product of a
coefficient, uniformly bounded on n and a function in L2([0, T ] : H−2(R)) uniformly bounded on n such
that v
t is bounded in L
2([0, T ] : H−2(R)). On the other hand, H1loc(R)
loc (R) →֒ H
−4(R). By
Lions-Aubin’s compactness Theorem [24] there is a subsequence v(nj) ≡ v(n) such that v(n) → v strongly
on L2([0, T ] : H
loc (R)). Hence, for a subsequence v
(nj) ≡ v(n), we have v(n) → v a. e. in L2([0, T ] :
loc (R)). Moreover, from (5.8), ∧v
5 ⇀ ∧v5 weakly in L
2([0, T ] : H−2(R)). Similarly, ∧v
2 ⇀ ∧v2
weakly in L2([0, T ] : H−2(R)). Since ||∧v(n)||H2(R) ≤ c ||v
(n)||L2(R) ≤ c ||v
(n)||H1(R) ≤ c ||v
(n)||H1/2(R) and
v(n) → v strongly on L2([0, T ] : H
loc (R)) then ∧v
(n) → ∧v strongly in L2([0, T ] : H2loc(R)). Thus, the
fifth term on the right hand side of (5.4), | ∧ v(n−1)|2 ∧ v
2 ⇀ | ∧ v|
2 ∧ v2 weakly in L
2([0, T ] : L1loc(R))
as ∧v
2 ⇀ ∧v2 weakly in L
2([0, T ] : H−2(R)) and | ∧v(n−1)|2 → |∧v|2 strongly on L2([0, T ] : H2loc(R)).
Similarly, the other terms in (5.4) converge to their limits, implying v
t ⇀ vt weakly in L
2([0, T ] :
L1loc(R)). Passing to the limit
i vt = ∂
2(i β ∧ v3 + ω ∧ v2 + | ∧ v|
2 ∧ v)− (i β ∧ v3 + ω ∧ v2 + | ∧ v|
2 ∧ v)
= −(I − ∂2)(i β ∧ v3 + ω ∧ v2 + | ∧ v|
2 ∧ v).
Thus i vt + (I − ∂
2)(i β ∧ v3 + ω ∧ v2 + | ∧ v|
2 ∧ v) = 0. This way, we have (2.4) for u = ∧v.
Now, we prove that there exists a solution of (2.4) with u ∈ L∞([0, T ] : HN (R)) and N ≥ 4, where T de-
pends only on the norm of u0 in H
3(R). We already know that there is a solution u ∈ L∞([0, T ] : H3(R)).
It is suffices to show that the approximating sequence v(n) is bounded in L∞([0, T ] : HN−2(R)). Taking
α = N − 2 and considering (5.5) for α ≥ 2, we define cN−2 =
||u0(·)||N + 1. Let T
N−3 be the largest
time such that ||v(k)( · , t)||α ≤ cN−3 for 0 ≤ t ≤ T
N−3, 0 ≤ k ≤ n. Integrating (5.5) over [0, t], for
0 ≤ t ≤ T
N−3, we have
ξ |v(n)α |
||v(n−1)||α
||v(n)||2αds+
||v(n−1)||α
It follows that
ξ(x, t) |v(n)α |
2dx ≤
ξ(x, 0) |v(n)α (x, 0)|
||v(n−1)||α
||v(n)||2αds
||v(n−1)||α
hence
|v(n)α |
2dx ≤
ξ |v(n)α |
2dx ≤
ξ(x, 0) |v(n)α (x, 0)|
||v(n−1)||α
||v(n)||2αds
||v(n−1)||α
|v(n)α |
2dx ≤
|v(n)α (x, 0)|
2dx +
G(cN−3)
c2N−3 t+
F (cN−3)
||v(n)α (x, 0)||
G(cN−3)
c2N−3 t+
F (cN−3)
||u(x, 0)||2N +
G(cN−3)
c2N−3 t+
F (cN−3)
and we obtain
||v(n)α ( · , t)||
αdx ≤
||u(x, 0)||2N +
G(cN−3)
c2N−3 t+
F (cN−3)
Claim. T
N−3 does not approach to 0.
On the contrary, assume that T
N−3 → 0. Since ||v
(n)( · , t)|| is continuous for t ≥ 0, there exists τ ∈
[0, TN−3] such that ||v
(k)( · , τ)||α = cN−3 for 0 ≤ τ ≤ T
(n), 0 ≤ k ≤ n. Then
c2N−3 ≤
||u(x, 0)||2N +
G(cN−3)
c2N−3 T
N−3 +
F (cN−3)
as n → +∞, and we have
||u(x, 0)||2N + 1
||u(x, 0)||2N then
4 γ21
||u(x, 0)||4N + 1 ≤ 0
which is a contradiction. Then T
N−3 6→ 0. By choosing TN−3 = TN−3(||u(x, 0)||
N ) sufficiently small, and
TN−3 not depending on n, we conclude that
||v(n)( · , t)||2α ≤ c
N−3 for all 0 ≤ t ≤ TN−3. (5.9)
This shows that T
N−3 ≥ TN−3. Thus,
v ∈ L∞([0, TN−3] : H
α(R)) ≡ L∞([0, TN−3] : H
N−2(R)).
Now, denote by 0 ≤ T ∗N−3 ≤ +∞ the maximal number such that for all 0 < t ≤ T
N−3, u = ∧v ∈
L∞([0, t] : HN (R)). In particular, TN−3 ≤ T
N−3 for all N ≥ 4. Thus, T can be chosen depending only
on the norm of u0 in H
3(R). Approximating u0 by {u
0 } ∈ C
0 (R) such that ||u0 − u
0 ||HN (R) → 0 as
j → +∞. Let uj be a solution of (2.4) with u(j)(x, 0) = u
0 . According to the above argument, there
exists T which is independent on n but depending only on supj ||u
0 || such that u
(j) there exists on [0, T ]
and a subsequence u(j)
−→ u in L∞([0, T ] : HN(R)).
As a consequence of Theorem 5.1 and 5.2 and its proof, one obtains the following result.
Corollary 5.3. Let |ω| < 3 β and let u0 ∈ H
N (R) with N ≥ 3 such that u
0 → u0 in H
N (R). Let
u and u(j) be the corresponding unique solutions given by Theorems 5.1 and 5.2 in L∞([0, T ] : HN(R))
with T depending only on supj ||u
0 ||H3(R) such that
⇀ u weakly on L∞([0, T ] : HN(R)),
u(j) → u strongly on L2([0, T ] : HN+1(R)).
6 Existence of Global Solutions
Here, we will try to extend the local solution u ∈ L∞([0, T ] : HN (W0 i 0)) of (2.4) obtained in Theo-
rem 5.2 to t ≥ 0. A standard way to obtain these extensions consists into deducing global estimations
for the HN (W0 i 0)-norm of u in terms of the H
N (W0 i 0)-norm of u(x, 0) = u0(x). These estimations
are frequently based on conservation laws which contain the L2-norm of the solution and their spatial
derivatives. It is not possible to do the same to give a solution of the problem of global existence because
the difficulty here is that the weight depends on the x and t variables. To solve our problem we follow a
different method using Leibniz’s rule like in the proof of Theorem 3.1 of Bona and Saut [3].
Theorem 6.1. For |ω| < 3 β there exists a global solution to (2.4) in the space Hs(R)∩HN (W0 i 0) with
N integer ≥ 3 and s ≥ 2.
Proof. The first part was proved in [3]. Differentiating (2.4) α-times (for α ≥ 0) over x ∈ R leads
i uα t + i β uα+3 + ω uα+2 + (|u|
2)α u+
(|u|2)α−m um + |u|
2 uα = 0. (6.1)
Let ξ = ξ(x, t), then multiplying (6.1) by ξ uα we have
i ξ uα uα t + i β ξ uα uα+3 + ω ξ uα uα+2 + (|u|
2)α ξ u uα
(|u|2)α−m ξ um uα + ξ |u|
2 |uα|
2 = 0
− i ξ uα uα t − i β ξ uα uα+3 + ω ξ uα uα+2 + (|u|
2)α ξ u uα
(|u|2)α−m ξ um uα + ξ |u|
2 |uα|
2 = 0. (applying conjugate)
Subtracting and integrating over x ∈ R we have
ξ |uα|
2dx+ i β
ξ uα uα+3dx+ i β
ξ uα uα+3dx+ ω
ξ uα uα+2dx (6.2)
ξ uα uα+2dx+ 2 i Im
ξ (|u|2)α u uαdx+ 2 i
ξ (|u|2)α−m um uαdx = 0.
Each term is calculated separately, integrating by parts in the second term we have
ξ uα uα+3dx =
∂2ξ uα uα+1dx+ 2
∂ξ |uα+1|
ξ uα+2 uα+1dx.
The other terms are calculated in a similar way. Hence in (6.2)
ξ |uα|
2dx− β
∂3ξ |uα|
2dx+ 3 β
∂ξ |uα+1|
2dx − 2ω Im
∂ξ uα uα+1dx
∂tξ |uα|
2dx+ 2 Im
ξ (|u|2)α u uαdx+ 2
ξ (|u|2)α−m um uαdx = 0
such that
ξ |uα|
2dx− β
∂3ξ |uα|
2dx+ 3 β
∂ξ |uα+1|
2dx+ 2 Im
(|u|2)α ξ u uαdx
∂tξ |uα|
2dx+ 2
ξ (|u|2)α−m um uαdx
= 2α Im
∂ξ uα uα+1dx ≤ |ω|
∂ξ |uα|
2dx+ |ω|
∂ξ |uα+1|
Hence
ξ |uα|
[ 3 β − |ω| ] ∂ξ |uα+1|
[ ∂tξ + β ∂
3ξ + |ω| ∂ξ ] |uα|
+ 2 Im
(|u|2)α ξ u uαdx+ 2
ξ (|u|2)α−mum uαdx ≤ 0. (6.3)
(|u|2)α = (u u )α =
uα−k uk = uuα +
uα−k uk + u uα
(|u|2)α u uα = |u|
2 |uα|
uα−k uk u uα + u
2 u2α
hence
( |u|2 )α ξ u uαdx = 2
ξ uα−k uk u uαdx+ 2 Im
ξ u2 u2αdx
ξ |uα−k| |uk| |u| |uα|dx+ 2
ξ |u|2 |uα|
ξ |uα−k| |uk| |u| |uα|dx+ 2 ||u||
L∞(R)
ξ |uα|
≤ 2 ||u||L∞(R)
ξ |uα−k| |uk| |uα|dx+ 2 ||u||
L∞(R)
ξ |uα|
2dx (6.4)
hence in (6.3) we have
ξ |uα|
[3 β − |ω| ] ∂ξ |uα+1|
2dx ≤
[∂tξ + β ∂
3ξ + |ω| ∂ξ + c ξ ] |uα|
ξ |uα−k| |uk| |u| |uα|dx− 2
ξ (|u|2)α−m um uαdx.
Using (2.3), Gagliardo-Nirenberg’s inequality and standard estimates we get
ξ |uα|
2dx+ [3 β − |ω| ]
∂ξ |uα+1|
2 dx ≤ c
ξ |uα|
2dx. (6.5)
Integrating (6.5) in t ∈ [0, Tmax = T ] we obtain
ξ |uα|
2dx+ [3 β − |ω| ]
∂ξ |uα+1|
2dx ds ≤ ||u0(x)||
ξ |uα|
where
ξ |uα|
2dx ≤ ||u0(x)||
ξ |uα|
Using Gronwall’s inequality
ξ |uα|
2dx ≤ ||u0(x)||
c t ≤ ||u0(x)||
it follows that
ξ |uα|
2dx ≤ c = c(T, ||u0(x)||
Then for any T = Tmax > 0 there exists c = c(T, ||u0(x)||
α) such that
||u||2α + [3 β − |ω| ]
∂ξ |uα+1|
2dx ds ≤ c.
This concludes the proof.
7 Persistence Theorem
As a starting point for the a priori gain of regularity results that will be discussed in the next section,
we need to develop some estimates for solutions of the equation (2.4) in weighted Sobolev norms. The
existence of these weighted estimates is often called the persistence of a property of the initial data u0.
We show that if u0 ∈ H
3(R)∩HL(W0 i 0) for L ≥ 0, i ≥ 1, then the solution u( · , t) evolves in H
L(W0 i 0)
for t ∈ [0, T ]. The time interval of that persistence is at least as long as the interval guaranteed by the
existence Theorem 5.2.
Theorem 7.1 (Persistence). Let |ω| < 3 β and let i ≥ 1 and L ≥ 0 be non-negative integers, 0 < T < +∞.
Assume that u is the solution to (2.4) in L∞([0, T ] : H3(R)) with initial data u0(x) = u(x, 0) ∈ H
3(R).
If u0(x) ∈ H
L(W0 i 0) then
u ∈ L∞
[0, T ] : H3(R) ∩HL(W0 i 0)
(7.1)
|∂L+1u(x, t)|2 η dx dt < +∞ (7.2)
where σ is arbitrary, η ∈ Wσ i 0 for i ≥ 1.
Proof. We use induction on α. Let
u ∈ L∞
[0, T ] : H3(R) ∩Hα(W0 i 0)
for 0 ≤ α ≤ L.
We derive formally some a priori estimate for the solution where the bound, involves only the norms of u
in L∞([0, T ] : H3(R)) and the norms of u0 in H
3(W0 i 0). We do this by approximating u(x, t) through
smooth solutions and the weight functions by smooth bounded functions. By Theorem 5.2, we have
u(x, t) ∈ L∞([0, T ] : HN (R)) with N = max{L, 3}.
In particular, uj(x, t) ∈ L
∞([0, T ]×R) for 0 ≤ j ≤ N − 1. To obtain (7.1) and (7.2) there are two ways
of approximation. We approximate general solutions by smooth solutions, and we approximate general
weight functions by bounded weight functions. The first of these procedure has already been discussed,
so we shall concentrate on the second.
Given a smooth weight function η(x) ∈ Wσ, i−1, 0 with σ > 0, we take a sequence η
ν(x) of smooth
bounded weight functions approximating η(x) from below, uniformly on any half line (−∞, c). Define
the weight functions for the α-th induction step as
(3 β − |ω|)
ην(y, t) dy
then the ξν are bounded weight functions which approximate a desired weight function ξ ∈ W0 i 0 from
below, uniformly on a compact set. For α = 0, multiplying (2.4) by ξν u, we have
i ξν uut + i β ξν uu3 + ω ξν u u2 + ξν |u|
4 = 0
− i ξν u ut − i β ξν u u3 + ω ξν u u2 + ξν |u|
4 = 0. (applying conjugate)
Subtracting and integrating over x ∈ R we have
ξν |u|
2dx − i
∂tξν |u|
2dx+ i β
ξν uu3dx+ i β
ξν u u3dx
ξν uu2dx− ω
ξν u u2dx = 0. (7.3)
Each term is treated separately, integrating by parts in the third term we have
ξν u u3dx =
∂2ξν uu1dx+ 2
∂ξν |u1|
ξν u2 u1dx.
The other terms are calculated in a similar way. Hence in (7.3) we have
ξν |u|
∂tξν |u|
2dx− β
∂3ξν |u|
2dx+ 3 β
∂ξν |u1|
= 2ω Im
∂ξν u u1dx ≤ |ω|
∂ξν |u|
2dx+ |ω|
∂ξν |u1|
Then, using (2.3) we obtain
ξν |u|
[3 β − |ω|] ∂ξν |u1|
[∂tξν + β ∂
3ξν + |ω| ∂ξν ] |u|
2dx ≤ c
ξν |u|
ξν |u|
2dx ≤ c
ξν |u|
We apply Gronwall’s Lemma to conclude that
ξν |u|
2dx ≤ c(T, ||u0||). (7.4)
for 0 ≤ t ≤ T, and c not depending on β > 0, the weighted estimate remains true for β → 0.
Now, we assume that the result is true for (α − 1) and we prove that it is true for α. To prove this, we
start from the main inequality (3.1) with ξ and η given by ξν and ην respectively.
ξν |uα|
ην |uα+1|
θν |uα|
Rαdx ≤ 0
where
ην = (3β − |ω| ) ∂ξν for |ω| < 3 β
θν = − [ ∂tξν + β ∂
3ξν + |ω| ∂ξν + c0 ξν ] where c0 = ||u||
L∞(R)
Rα = Rα(|uα|, |uα−1|, . . . )
ξν |uα|
2dx +
ην |uα+1|
2dx ≤ −
θν |uα|
2dx −
∣∣∣∣−
θν |uα|
∣∣∣∣ ≤
|θν | |uα|
|Rα|dx.
Using (2.3) in the first part of the right hand side we obtain
θν |uα|
2dx ≤ c
ξν |uα|
ξν |uα|
ην |uα+1|
2dx ≤ c
ξν |uα|
|Rα|dx. (7.5)
According to (3.8),
Rα dx contains a term of the form
ξν uν1 uν2 uαdx. (7.6)
We estimate the term
ξν uν1 uν2 uα dx for ν1 + ν2 = α. (7.7)
Let ν2 ≤ α− 2. Integrating by parts one time in (7.7) we have
ξν uν1 uν2 uα dx = −
∂ξν uν1 uν2 uα−1 dx−
ξν uν1+1 uν2 uα−1 dx
ξν uν1 uν2+1 uα−1 dx.
We estimates the first term in the right hand side in (7.7). Using Holder’s inequality and standard
estimates we obtain
ξν |uν2+1|
ξν |uν2 |
)1/2](∫
ξν |uα−1|
(7.8)
where (7.8) is bounded by hypothesis. The other terms are estimates in a similar way. Now suppose that
ν1 = ν2 = α− 1, then in (7.7) we have
ξν uα−1 uα−1 uαdx,
hence
ξν |uα−1|
2 uαdx
∣∣∣∣ ≤ ||uα−1||L∞(R)
ξν |uα−1|
)1/2 (∫
ξν |uα|
where ||uα−1||L∞(R) is bounded by hypothesis, and the estimate is complete. In a similar way we estimate
all the other terms of Rα. Using these estimates in (7.5) and applying Gronwall’s argument, we obtain
for 0 ≤ t ≤ T
ξν |uα|
ην |uα+1|
2dx ≤ c0 e
ξν |∂
αu0(x)|
2dx+ 1
where c0 and c1 are independent of ν and such that letting the parameter ν → 0 the desired estimate
(7.2) is obtained.
8 Main Theorem
In this section we state and prove our main theorem, which states that if the initial data u(x, 0) decays
faster than polynomially on R+ = {x ∈ R : x > 0} and possesses certain initial Sobolev regularity, then
the solution u(x, t) ∈ C∞ for all t > 0.
If η is an arbitrary weight function in Wσ i k, then by Lemma 3.2, there exists ξ ∈ Wσ, i+1, k which
satisfies (3.1). For the main theorem, we take 4 ≤ α ≤ L+ 2. For α ≤ L+ 4, we take
η ∈ Wσ, L−α−2, α−3 =⇒ ξ ∈ Wσ, L−α−3, α−3. (8.1)
Lemma 8.1(Estimate of error terms). Let 4 ≤ α ≤ L+2 and the weight functions be chosen as in (8.1),
∣∣∣∣∣
(θ |uα|
2 +Rα)dx dt
∣∣∣∣∣
≤ c, (8.2)
where c depends only on the norms of u in
L∞([0, T ] : Hβ(Wσ, L−β+3, β−3)) ∩ L
2([0, T ] : Hβ+1(Wσ, L−β+2, β−3))
for 3 ≤ β ≤ α− 1, and the norms of u in L∞([0, T ] : H3(W0 L 0)).
Proof. We must estimate both Rα and θ. We begin with a term in Rα of the form
ξ |uν1 | |uν2 | |uα| (8.3)
assuming that ν1 ≤ α− 2.
By the induction hypothesis, u is bounded in L∞([0, T ] : Hβ(Wσ, L−(β−3)+, (β−3)+)) for 0 ≤ β ≤ α − 1.
By Lemma 2.1,
ζ |uβ|
2 < +∞ (8.4)
for 0 ≤ β ≤ α − 2 and ζ ∈ Wσ, L−(β−2)+, (β−2)+ . We estimate |uν1 | using (8.4). We estimate |uν2 | and
|uα| using the weighted L
2 bounds
ζ |uν2 |
2dx dt < +∞ for ζ ∈ Wσ, L−(ν2−3)+, (ν2−4)+ (8.5)
and the same with ν2 replaced by α. It suffices to check the powers to t, the powers of x as x → +∞ and
the exponential of x as x → −∞.
For x > 1. In the (8.3) term, the factor ξ contributed according to (8.1)
ξ(x, t) = tα−3 x(L−α+3) t−(α−3) x−(L−α+3)ξ(x, t) ≤ c2 t
α−3 x(L−α+3) (using(2.3))
then ξ |uν1 | |uν2 | |uα| ≤ c2 t
α−3 x(L−α+3)|uν1 | |uν2 | |uα|. Moreover
|uν1 | |uν2 | |uα| = t
(ν1−2)
L−(ν1−2)
−(ν1−2)
(L−(ν1−2)
2 |uν1 | ×
(ν2−4)
L−(ν2−3)
−(ν2−4)
(L−(ν2−3)
2 |uν2 | ×
(α−4)+
L−(α−3)+
−(α−4)+
(L−(α−3)+)
2 |uα|.
tt follows that
ξ |uν1 | |uν2 | |uα|
≤ c2 t
M xT t
(ν1−2)
L−(ν1−2)
2 |uν1 | t
(ν2−4)
L−(ν2−3)
2 |uν2 | t
(α−4)+
L−(α−3)+
2 |uα| (8.6)
where
M = α− 3−
(ν1 − 2)
(ν2 − 4)
(α− 4)+
T = (L− α+ 3)−
(L − (α− 3)+)−
(L− (ν2 − 3)
(L − ν1 − 2)
Claim. M ≥ 0 is large enough, that the extra power of t can be omitted
2M = 2α− 6− (ν1 − 2)
+ − (ν2 − 4)
+ − (α− 4)+
= α− 2− (ν1 − 2)
+ − (ν2 − 4)
= α− 2− ν1 + 2− ν2 + 4 = α+ 4− (ν1 + ν2)
= α+ 4− α = 4 ≥ 0.
Claim. T ≤ 0 is such that the extra power of t can be omitted.
2T = 2L− 2α+ 6− L+ (α− 3)+ − L+ (ν2 − 3)
+ − L+ (ν1 − 2)
= −L− α+ ν1 + ν2 − 2 = −L− α+ α− 2
= −(L+ 2) ≤ 0.
Now, we study the behavior as x → −∞. Since each factor uνj (j = 1, 2) must grow slower that an
exponential eσ
′ |x| and ξ decays as an exponential e−σ |x|, we simply need to choose the appropriate rela-
tionship σ and σ′ at each induction step. The analysis will be completed with the case where ν1 ≥ α− 1.
Then, in (3.9), if 2(α − 1) ≤ α, but α ≥ 3. So this possibility is impossible. For x < 1 the estimate is
similar, except for an exponential weight. The analysis of all terms of Rα is estimated in a similar form.
This completes the estimate of Rα.
Now, we estimate the term θ |uα|
2 where θ is given in (3.1). We have that θ involves derivatives of u only
up to order one, and hence, θ |uα|
2 is a sum of terms of the same type which we have already encountered
in Rα. So, its integral can be bounded in the same type. Indeed, (3.1) shows that θ depends on ξt, ∂
and derivatives of lower order. By using (3.6) we have the claim.
Theorem 8.2(Main Theorem). Let |ω| < 3 β, T > 0 and u(x, t) be a solution of (2.4) in the region
R× [0, T ] such that
u ∈ L∞([0, T ] : H3(W0 L 0)) (8.7)
for some L ≥ 2. Then
u ∈ L∞([0, T ] : H3+l(Wσ, L−l, l)) ∩ L
2([0, T ] : H4+l(Wσ, L−l−1, l)) (8.8)
for all 0 ≤ l ≤ L− 1 and all σ > 0.
Remark. If the assumption (8.7) holds for all L ≥ 2, the solution is infinitely differentiable in the x-
variable. ¿From (2.4) we have that the solution is C∞ in both variables. We are also quantifying the
gain of each derivative by the degree of vanishing of the initial data at infinity.
Proof. We use induction on α. For α = 3, let u be a solution of (2.4) satisfying (8.7). Therefore,
ut ∈ L
∞([0, T ] : L2(W0 L 0)) where u ∈ L
∞([0, T ] : H3(W0 L 0)) and ut ∈ L
∞([0, T ] : L2(W0 L 0)).
Then u ∈ C([0, T ] : L2(W0 L 0)) ∩ Cw([0, T ] : H
3(W0 L 0)). Hence, u : [0, T ] 7−→ H
3(W0 L 0) is
a weakly continuous function. In particular, u( · , t) ∈ H3(W0 L 0) for all t. Let t0 ∈ (0, T ) and
u( · , t0) ∈ H
3(W0 L 0), then there are {u
0 } ⊆ C
0 (R) such that u
0 ( · ) → u( · , t0) in H
3(W0 L 0).
Let u(n)(x, t) be a unique solution of (2.4) with u(n)(x, t0) = u
0 . Then by Theorem 5.1 and 5.2, there
exists u in a time interval [t0, t0 + δ] where δ > 0 does not depend on n and u is a unique solution of
(2.4), u(n) ∈ L∞([t0, t0 + δ] : H
3(W0 L 0)) with u
(n)(x, t0) ≡ u
0 (x) → u(x, t0) ≡ u0(x) in H
3(W0 L 0).
Now, by Theorem 7.1, we have
u(n) ∈ L∞([t0, t0 + δ] : H
3(W0 L 0)) ∩ L
2([t0, t0 + δ] : H
4(Wσ, L−1, 0))
with a bound that depends only on the norm of u
0 in H
3(W0 L 0). Furthermore, Theorem 7.1 guarantees
the non-uniform bounds
[t0, t0+δ]
(1 + |x+|)
k | ∂αu(n)(x, t) | < +∞
for each n, k and α. The main inequality (3.1) and the estimate (8.2) are therefore valid for each u(n) in
the interval [t0, t0 + δ]. η may be chosen arbitrarily in its weight class (8.1) and then ξ is defined by (3.7)
and the constant c1, c2, c3, c4 are independent of n. From (3.1) and (8.1) we have
[t0, t0+δ]
ξ |u(n)α |
∫ t0+δ
2dx ≤ c (8.9)
where by (8.2), c is independent of n. The estimate (8.9) is proved by induction for α = 3, 4, 5, . . . Thus
u(n) is also bounded in
L∞([t0, t0 + δ] : H
α(Wσ, L−α+3, α−3)) ∩ L
2([t0, t0 + δ] : H
α+1(Wσ, L−α+2, α−3)) (8.10)
for α ≥ 3. Since u(n) → u in L∞([t0, t0 + δ] : H
3(W0 L 0)). By Corollary 5.3 it follows that u belongs to
the space (8.10). Since δ is fixed, this result is valid over the whole interval [0, T ].
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	Introduction
	Preliminaries
	Main Inequality
	An a priori estimate
	Uniqueness and Existence of a Local Solution
	Existence of Global Solutions
	Persistence Theorem
	Main Theorem
