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On the initial value problem of the semilinear beam equation with weak damping I: Smoothing effect
J. Math. Anal. Appl. (
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On the initial value problem of the semilinear beam equation with weak damping I: Smoothing effect Hiroshi Takeda a,∗ , Shuji Yoshikawa b a
Department of Intelligent Mechanical Engineering, Faculty of Engineering, Fukuoka Institute of Technology, 3-30-1 Wajiro-higashi, Higashi-ku, Fukuoka, 811-0295, Japan b
Department of Engineering for Production and Environment, Graduate School of Science and Engineering, Ehime University, 3 Bunkyo-cho, Matsuyama, Ehime, 790-8577, Japan
article
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Article history: Received 15 March 2012 Available online xxxx Submitted by David Russell Keywords: Damped beam equation Cauchy problem Fourth order wave equation with damping Asymptotic profile Perturbation
This paper is devoted to the study of the initial value problem for some semilinear damped beam equation. We shall prove the unique global existence of a small solution for the equation and also obtain a certain smoothing effect of the solution. Our method is based on the expansion formula for the propagator to decompose into the dissipative part and the dispersive part. To control the dispersive part without regularity loss, we apply the stationary phase method developed by Van der Corput. As a result, we observe the smoothing effect of the solution which does not occur on the solution for damped wave equations. © 2012 Elsevier Inc. All rights reserved.
1. Introduction In this paper we study the initial value problem for the following semilinear damped beam equation
∂t2 u + ∂t u + ∂x4 u − α∂x2 u = ∂x f (∂x u), u(0, x) = g0 (x),
∂t u(0, x) = g1 (x),
t > 0, x ∈ R, x ∈ R,
(1.1) (1.2)
where u = u(t , x); (0, T )× R → R is an unknown function, g0 (x) and g1 (x) are given initial data and α is a positive constant. We assume that the nonlinear function f ∈ C 1 (R) satisfies that for some p ≥ 2 and small any v , v1 and v2 ∈ R,
|f (v)| ≤ C |v|p ,
|f ′ (v)| ≤ C |v|p−1 , |f (v1 ) − f (v2 )| ≤ C |v1 |p−1 + |v2 |p−1 |v1 − v2 |, |f ′ (v1 ) − f ′ (v2 )| ≤ C |v1 |p−2 + |v2 |p−2 |v1 − v2 |,
(1.3)
where C is independent of v , v1 and v2 . The typical example is given by f (v) = −|v|p−1 v,
p ≥ 2.
The lowest exponent 2 for p is needed for all the powers in (1.3) to be non-negative.
∗
Corresponding author. E-mail addresses: [email protected] (H. Takeda), [email protected] (S. Yoshikawa).
0022-247X/$ – see front matter © 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2012.12.015
(1.4)
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Our aim of this paper is to show the existence of a global solution u ∈ C ([0, ∞); L1 (R) ∩ H 2 (R)) of (1.1)–(1.2) for small data (g0 , g1 ) ∈ W 2,1 (R) ∩ H 2 (R) × L1 (R) ∩ L2 (R) that has the decay rates
∥u(t )∥
Lq (R)
− 21 1− 1q
≤ C (1 + t )
∥∂x2 u(t )∥Lq (R) ≤ Ct
− 12 1− 1q
−1
,
t ≥ 0, 1 ≤ q ≤ ∞,
,
t > 0, 2 ≤ q ≤ ∞.
(1.5) (1.6)
According to the result in [10,23], the solution of the initial value problem for the linear equation
∂t2 u + ∂t u + (−∂x2 )ℓ u = 0,
t > 0, x ∈ R, ℓ ∈ N
decays − 21ℓ 1− 1q
∥u(t )∥Lq (R) ≤ C (1 + t )
,
t ≥ 0, 1 ≤ q ≤ ∞.
Then the decay estimate (1.5) coincides with that of the damped wave equation
∂t2 u + ∂t u − ∂x2 u = 0,
t > 0, x ∈ R,
(1.7)
which means that the decay property of the solution to (1.1)–(1.2) is mainly dominated by the lower order term α∂x2 u (α > 0). For the corresponding nonlinear problem to (1.1)–(1.2) in α = 0, we refer to [23,24]. The decay estimate (1.6) implies that the solution also belongs to the class C ((0, ∞); W 2,∞ (R)). Such a smoothing effect arises from the dispersive property and never occurs on the solution for damped wave equations under the above data. Moreover, in the sequel [25], we shall give several asymptotic profiles of the decaying solution. As a consequence, we see that the decay rate (1.5) and (1.6) are optimal. Before discussing the result of this paper more precisely, we shall explain the background of our problem and the related results. The linear beam equation with damping
∂t2 u + (−∂x2 )m ∂t u + ∂x4 u = 0 is studied by many authors (see [14] and reference therein). In particular, the equation is called weak damped type when m = 0. Under the constant distributed temperature T higher than the critical temperature Tc of martensitic phase transitions in shape memory alloys, the Falk model for the displacement u is written as
∂t2 u + ∂x4 u − α∂x2 u = ∂x f (∂x u), where α = a1 (T − Tc ) and f (v) = −a2 |v|p2 −1 v + a3 |v|p3 −1 v
(1.8)
for positive physical constants a1 , a2 and a3 with p2 = 3 and p3 = 5. We refer to the monograph [2, Chapter 5] for more precise information and the derivation of the Falk model. The initial boundary value problem of the equation with weak damping (i.e. (1.1) with (1.8)) was studied in [21]. They obtained existence of time global solution with large data and its global attractor in bounded spatial domain. It plays an essential role in their proof that F (v) = a3 |v|p3 +1 /(p3 + 1) − a2 |v|p2 +1 /(p2 + 1) is bounded from below and the boundedness of the spatial domain. The double-well form (1.8) also appears in the Cahn–Hilliard equation
∂t u + ∆2 u = ∆f (u).
(1.9)
Evans–Galaktionov–Williams [4] studied the Cahn–Hilliard equation (1.9) of limit case as a3 → 0 called limit unstable case. On the other hand, they called the case a2 → 0 limit stable case. To sum up, for f (v) = a|v|p−1 v , a > 0 and a < 0 are regarded as the limit stable case and the limit unstable case, respectively. Grasselli et al. in [6,5] considered the Cahn–Hilliard equation with inertial term
∂t2 u + ∂t u + ∆2 u = −∆f (u),
t > 0, x ∈ Ω (b Rn , n = 2, 3).
In [24] authors studied the Cauchy problem of limit unstable Cahn–Hilliard equation with inertial term
∂t2 u + ∂t u + ∆2 u = −∆(|u|p−2 u),
t > 0 , x ∈ Rn ,
(1.10)
for n = 1, 2, 3. As mentioned above, the solution for (1.10) decays − 4n 1− 1q
∥u(t )∥Lq (Rn ) ≤ C (1 + t )
,
t ≥ 0, 2 ≤ q ≤ ∞.
We can observe that the rate of the decay estimate is slower than (1.5), due to the absence of the second order term α ∆u. Moreover, we note that in this article we also obtain the smoothing estimate (1.6) restricting ourselves to the onedimensional case. The same strategy does not work in the multi-dimensional case. Correspondingly, for the semilinear damped wave equation
∂t2 u + ∂t u − ∆u = a|u|p−1 u,
t > 0, x ∈ Rn ,
(1.11)
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the right hand side is called the absorbing term when a < 0 and the sourcing term when a > 0. These terms play completely different roles and these analyses are also different. Indeed, making use of the energy method, the absorbing type (1.11) with a < 0 have a solution without the smallness assumption of the data, and further the decay estimate for the solution can be obtained (see e.g. [7,11,19]). On the other hand, in the case of the sourcing type with a > 0, for a certain small p the blow-up result was shown by Nishihara [17] (we also refer to [13,28]). It is also known that the small data global existence and the asymptotic profile of the solution to (1.11) for a certain large p is given by the heat kernel (see e.g. [8,9,20] and reference therein). The Eq. (1.1) with (1.4) corresponds to the limit unstable case of the Falk model with weak damping. From the mathematical point of view the limit stable case and limit unstable case of (1.1) correspond to the absorbing type and the sourcing type of (1.11), respectively. For (1.1) of the limit stable case, authors have shown that there exists time global solution with decay for large data in [26]. In this study we are interested in the Eq. (1.1) of the limit unstable case (i.e. (1.1) with (1.4)). For the problem the large data global existence is no longer expected. Let us state our main results of this paper. We call u(t ) a mild solution of the initial value problem (1.1)–(1.2) if u(t ) ∈ C ([0, ∞); L1 (R) ∩ H 2 (R)) and satisfies the integral equation u(t ) = K0 (t )g0 + K1 (t )
1 2
g0 + g1
t
+
K1 (t − s)∂x f (∂x u(s))ds,
(1.12)
0
where the evolution operators K0 (t ) and K1 (t ) are given by
K0 (t )f = F
−1
− 2t
e
it
ξ 4 +αξ 2 − 14
+e
−it ξ 4 +αξ 2 − 14
2
K1 (t )g = F −1 e
e
− 2t
e
it
ξ 4 +αξ 2 − 14
−e
−it ξ 4 +αξ 2 − 41
2i ξ 4 + αξ 2 −
1 4
f,
(1.13)
g .
(1.14)
The following result is the global existence of a mild solution for small data with the decay. Theorem 1.1 (Unique Global Existence of Solution). Assume that f satisfies the assumption (1.3) for p ≥ 2. If the initial data
(g0 , g1 ) ∈ W 2,1 (R) ∩ H 2 (R) × L1 (R) ∩ L2 (R) and ∥g0 ∥W 2,1 (R)∩H 2 (R) + ∥g1 ∥L1 (R)∩L2 (R)
is sufficiently small, then there exists a unique mild solution u(t ) for the initial value problem (1.1)–(1.2) in C ([0, ∞); L1 (R) ∩ H 2 (R)) satisfying
∥u(t )∥L1 (R) ≤ C ,
5
∥∂x2 u(t )∥L2 (R) ≤ C (1 + t )− 4 ,
t ≥ 0.
(1.15)
Remark 1.2. Theorem 1.1 states the existence of a kind of weak solutions for (1.1)–(1.2). On the other hand, we can prove the existence of a solution in C ([0, ∞); W s,1 (R) ∩ H s+2 (R)) for s ≥ 0 under the corresponding additional regularity assumption of the data and the nonlinearity f , by the same procedure as the proof of Theorem 1.1. Our next purpose is to show the smoothing effect of the global in time solution for (1.1)–(1.2) constructed in Theorem 1.1. Theorem 1.3 (Smoothing Effect). Under the same condition as in Theorem 1.1, the mild solution u(t ) to (1.1)–(1.2) satisfies u(t ) ∈ C ([0, ∞); L1 (R) ∩ H 2 (R)) ∩ C ((0, ∞); W 2,∞ (R)). Moreover, the following time decay estimate holds
∥∂x2 u(t )∥Lq (R) ≤ Ct
− 21 1− 1q −1
,
2 ≤ q ≤ ∞, t > 0.
(1.16)
In Theorem 1.3, ‘‘smoothing effect’’ means the time global decay of the solution in the sense of W 2,∞ for the initial data (g0 , g1 ) ∈ W 2,1 ∩ H 2 × L1 ∩ L2 . The smoothing arises from the dispersive property of the fourth order term, since the estimate (1.16) does not hold for the solution of the damped wave equation (1.7) under the same assumption in Theorem 1.3. Our strategy for the proof of theorems lies in the application of the Lq –Lr estimates for the linearized solution of the damped beam equation. To obtain the estimates, we use the Fourier splitting method and decompose the linearized solution into dissipative and dispersive parts.
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Nishihara in [18] (see also [9,15,16]) showed that for the solutions of linear heat and wave equations
∂t v − ∆v = 0, v(0, x) = g0 (x) + g1 (x),
t > 0 , x ∈ Rn , x ∈ Rn ,
2 ∂t w − ∆w = 0, w(0, x) = g0 (x), ∂t w(0, x) = g1 (x),
t > 0, x ∈ Rn , x ∈ Rn ,
the solution of the damped wave equation (1.7) can be decomposed into v(t ) and w(t ) such as t
u(t ) = v(t ) + e− 2 w(t ) + error
as t → ∞.
(1.17)
This expansion (1.17) states that the factor of w(t ) in the solution is asymptotically small when t is large. In contrast, we
established the decomposition of K1 (t ) and its second order derivative ∂x2 K1 (t ) into the dissipative evolution et α∂x and the 2
t
dispersive part e− 2 e±it ∂x such as 2
t
K1 (t ) = et α∂x + e− 2 ∂x−2 e±it ∂x + error 2
∂
2 x K1
2 t α∂x2 xe
(t ) = ∂
t
2
+e
− 2t ±it ∂x2
e
as t → ∞,
(1.18)
+ error as t → ∞,
where e− 2 e±it ∂x decays faster than the dissipative part. Indeed, the decay order of the solution is dominated by the dissipative part without ‘‘dispersive singularities’’ and we can obtain the decay estimates as the solution of the linear heat equation. On the other hand, we need to pay attention to the dispersive part, since the regularity property of the solution mainly depends on the high frequency part, in general. Then our analysis for the dispersive part, which is based on the stationary phase method, makes us possible to control the L∞ -norm of ∂x2 K1 (t ) without the additional regularity assumption for the data. Summing up the above, the expansion (1.18) enables us to obtain the smoothing property and the asymptotic profile for our global solution constructed in Theorem 1.1. We conclude the introduction by giving notation used in this paper. We denote by ∂y partial differential operator with respect to a variable y. Let W k,p (R) be the Sobolev space for k ∈ N and 1 ≤ p ≤ ∞ defined by 2
W k,p (R) = {u : R → R; ∥u∥W k,p (R) = ∥u∥Lp (R) + ∥∂xk u∥Lp (R) < ∞}, and H k (R) = W k,2 (R). We denote the Fourier and the Fourier inverse transforms by F and F −1 . We also let f the Fourier transform of f . We denote the several positive constants by C > 0 without confusions. These constants may change from line to line. 2. Preliminaries For f ∈ L2 (Rn ) ∩ Lq (Rn ), 1 ≤ q ≤ ∞, let m(ξ ) be the Fourier multiplier defined by n
F −1 [m f ](x) = (2π )− 2
Rn
eix·ξ m(ξ ) f (ξ )dξ .
We define Mq as the class of the Fourier multiplier for 1 ≤ q ≤ ∞:
Mq =
m : Rn → R, there exists a positive constant Cq < ∞ such that ∥F −1 [m f ]∥Lq (Rn ) ≤ Cq ∥f ∥Lq (Rn ) ,
and for m ∈ Mq , we let Mq (m) = sup f ̸=0
∥F −1 [m f ]∥Lq (Rn ) . q ∥f ∥L (Rn )
The following lemma is known as the Carleson–Beurling inequality, which is applied to show the Lq –Lq boundedness of the Fourier multiplier. Lemma 2.1 (The Carleson–Beurling Inequality [1]). If m ∈ H s (Rn ) with s > following estimate holds 1− n
n , 2
then m ∈ Mq for all 1 ≤ q ≤ ∞. Moreover, the
n
Mq (m) ≤ C ∥m∥L2 (R2sn ) ∥∂ξs m∥L2s2 (Rn ) . We introduce the following classical lemma by Van der Corput to control the oscillation integral without the additional regularity assumption.
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Lemma 2.2 ([12, Lemma 2.8]). Assume φ ∈ C 2 ([a, b]), |φ ′′ (ξ )| > 1 for ξ ∈ [a, b]. If |ψ(b)| < ∞ and ψ ′ ∈ L1 (a, b), then,
b
e
it φ(ξ )
a
− 12 |ψ(b)| + ψ(ξ )dξ ≤ 10|t |
b
|ψ (ξ )|dξ . ′
a
To compute the decay order of the linear estimates, we make an extensive use of the following well-known estimates. Lemma 2.3. For d ∈ N ∪ {0} and positive constant β , there exists a constant C depending only on d and β such that
1
|ξ |d e−β(1+t )|ξ | dξ ≤ C (1 + t )− 2 (1+d) . 2
R
The next lemma is useful to compute the decay order of nonlinear part in the integral equation (1.12). Lemma 2.4. Given any t ≥ 0, the following properties hold:
(i) Let a > 0 and b > 0 with min{a, b} > 1. There exists a constant C depending only on a and b such that t (1 + t − s)−a (1 + s)−b ds ≤ C (1 + t )− min{a,b} .
(2.1)
0
(ii) Let 1 > a ≥ 0, b > 0 and c > 0. There exists a constant C independent of t such that t e−c (t −s) (t − s)−a (1 + s)−b ds ≤ C (1 + t )−b .
(2.2)
0
For the proof of (2.1), we refer to Segal [22]. The estimate (2.2) is well-known (see e.g. Yamada [27]). 3. Linear estimates In this section we introduce several linear estimates. Thereafter we use the cut-off functions which will be used in the proofs to aligned to low-, middle- and high-frequency parts. Let χl , χm and χh ∈ C ∞ (R) be
ρ |ξ | ≤ , 1, χ (ξ ) = 2 h 0, 0, |ξ | ≥ ρ, χm (ξ ) = 1 − χl (ξ ) − χh (ξ )
χl (ξ ) =
1,
|ξ | ≥ 2R, |ξ | ≤ R,
for ρ ≪ 1 and R ≫ 1 which will be determined in each proof. We denote by δ < 1/2 a positive constant which may change from line to line. First, we show the dissipative property for K0 (t ) and K1 (t ) defined in (1.13) and (1.14). We remark that the following proposition holds in q < 2. For the solution of the damped wave equation such estimates were firstly obtained in [18]. Here, we apply the method developed by Hosono–Ogawa [9] using the Carleson–Beurling inequality. Proposition 3.1 (Lq –Lr Estimates). For any 1 ≤ r ≤ q ≤ ∞ and t ≥ 0 we have − 21
1 1 r −q
− 21
1 1 r −q
∥K0 (t )g ∥Lq (R) ≤ C (1 + t )
∥K1 (t )g ∥Lq (R) ≤ C (1 + t )
∥g ∥W 2,r (R) ,
(3.1)
∥g ∥Lr (R) .
(3.2)
Proof. We define the Fourier multiplier m1 (t , ξ ) corresponding to the evolution operator K1 (t ) by m1 ( t , ξ ) = e
− 2t
e
it
ξ 4 +αξ 2 − 14
−e
−it ξ 4 +αξ 2 − 14
2i ξ 4 + αξ 2 −
1 4
.
If we show 1
∥K1 (t )g ∥L∞ (R) ≤ C (1 + t )− 2 ∥g ∥L1 (R)
(3.3)
∥K1 (t )g ∥Lq (R) ≤ C ∥g ∥Lq (R) ,
(3.4)
and 1 ≤ q ≤ ∞,
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then we obtain (3.2) from the interpolation. We first show L∞ –L1 estimate (3.3). It follows from the Hausdorff–Young inequality that
∥K1 (t )g ∥L∞ (R) ≤ C ∥m1 (t , ·) g ∥L1 (R) ≤ C ∥m1 (t , ·)∥L1 (R) ∥ g ∥L∞ (R) ≤ C ∥m1 (t , ·)∥L1 (R) ∥g ∥L1 (R) . √ 1+α 2 2
Setting ζ (α) = − α2 +
(> 0), we see
4 ξ + αξ 2 − 1/4 ≤ 0, ξ 4 + αξ 2 − 1/4 > 0,
|ξ | ≤ ζ (α), |ξ | > ζ (α).
Then, we should regard as
1 4 − αξ 2 sinh − ξ t 4 t , e− 2 1 4 − αξ 2 − ξ 4 m1 (t , ξ ) = sin t ξ 4 + αξ 2 − 41 t −2 e , ξ 4 + αξ 2 − 1
|ξ | < ζ (α), (3.5)
|ξ | > ζ (α).
4
For a small constant ρ < ζ (α), we define the positive constant by Cρ,α =
α 1 2
+
1 4
− ρ 4 − αρ 2
.
(3.6)
We see that for |ξ | < ρ
t 14 −ξ 4 −αξ 2 −t 14 −ξ 4 −αξ 2 t sinh t 14 − ξ 4 − αξ 2 − e t e −2 e ≤ e− 2 1 4 − αξ 2 2 14 − ξ 4 − αξ 2 − ξ 4 1
1
2
4
≤ C exp t − +
− ξ 4 − αξ 2
1
≤ C exp (1 + t ) − + 2
1 4
− ξ 4 − αξ 2
≤ C exp{−Cρ,α (1 + t )ξ 2 },
(3.7)
where in the last inequality we used the fact that 1
− + 2
1 4
− ξ 4 − αξ 2 =
−(ξ 4 + αξ 2 ) ≤ + 14 − ξ 4 − αξ 2
1 2
−αξ 2 1 2
+
1 4
− ρ 4 − αρ 2
.
Therefore, by Lemma 2.3 we obtain
∥χl m1 (t , ·)∥L1 (R) ≤ C
e−Cρ,α (1+t )ξ dξ 2
|ξ |≤ρ 1
≤ C (1 + t )− 2 . Now we fix some large R > ζ (α). Then it follows that
∥χm m1 (t , ·)∥L1 (R) ≤ C
ρ/2≤|ξ |≤2R
|m1 (t , ξ )|dξ ≤ Ce−δt
(3.8)
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from
1 4 − αξ 2 sinh t − ξ 4 t , te− 2 1 t 4 − ξ 4 − αξ 2 |m1 (t , ξ )| ≤ t sin t ξ 4 + αξ 2 − 41 − , te 2 t ξ 4 + αξ 2 − 41
(ρ/2 ≤ |ξ | < ζ (α)),
(ζ (α) < |ξ | ≤ 2R)
t
≤ Cte− 2 ≤ Ce−δt . It also follows that
∥χh m1 (t , ·)∥L1 (R) ≤ C
t
|m1 (t , ξ )|dξ ≤ Ce− 2 ,
(3.9)
|ξ |≥R
from
sin t ξ 4 + αξ 2 − 14 1 ≤ ξ 4 + αξ 2 − 41 ξ 4 + αξ 2 −
∈ L1 (|ξ | ≥ R).
1 4
Consequently, we have
∥m1 (t , ·)∥L1 (R) ≤ ∥χl m1 (t , ·)∥L1 (R) + ∥χm m1 (t , ·)∥L1 (R) + ∥χh m1 (t , ·)∥L1 (R) 1
≤ C (1 + t )− 2 . Next, we verify Lq –Lq estimate (3.4). Recall the estimate (3.7) and Lemma 2.3, then we have
∥χl m1 (t , ·)∥L2 (R) ≤ C
e−2Cρ,α (1+t )ξ dξ 2
21
|ξ |≤ρ 1
≤ C (1 + t )− 4 . Similar computations to (3.8) and (3.9) also yield
∥χm m1 (t , ·)∥L2 (R) + ∥χh m1 (t , ·)∥L2 (R) ≤ Ce−δt .
(3.10)
Then we deduce that
∥m1 (t , ·)∥L2 (R) ≤ ∥χl m1 (t , ·)∥L2 (R) + ∥χm m1 (t , ·)∥L2 (R) + ∥χh m1 (t , ·)∥L2 (R) 1
≤ C (1 + t )− 4 .
(3.11)
We note that for |ξ | < |ζ (α)| − 2t
∂ξ {χ (ξ )m1 (t , ξ )} = −te
− 2t
+e
cosh t
1 4
sinh t
1 4
2ξ 3 + αξ
− ξ 4 − αξ 2
− ξ 4 − αξ 2
1 4
− ξ 4 − αξ 2
2ξ 3 + αξ
1 4
t
+ e− 2
sinh t
1 4
1 4
− ξ − αξ 4
2
− ξ 4 − αξ 2
χ (ξ ) 32 χ (ξ )
χ ′ (ξ ),
− ξ 4 − αξ 2
and for |ζ (α)| > |ξ |
∂ξ {χ (ξ )m1 (t , ξ )} = −te
− 2t
− 2t
+e
cos t
ξ + αξ − 4
2
sin t
ξ + αξ − 4
2
1
4 1 4
2ξ 3 + αξ
ξ 4 + αξ 2 −
1 4
2ξ 3 + αξ
ξ 4 + αξ 2 −
1 4
χ (ξ ) 32 χ (ξ )
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sin t
t
+ e− 2
ξ 4 + αξ 2 −
ξ + αξ − 4
2
)
–
1 4
χ ′ (ξ ).
1 4
We shall use the above expression of ∂ξ {χ (ξ )m1 (t , ξ )} by replacing χ to χl , χm and χh . Recalling ρ < ζ (α), from (3.7) we have for |ξ | ≤ ρ
|∂ξ {χl (ξ )m1 (t , ξ )}| ≤ Ct |ξ |e−Cρ,α (1+t )ξ χl (ξ ) + C |ξ |e−Cρ,α (1+t )ξ χl (ξ ) + Ce−Cρ,α (1+t )ξ |χl′ (ξ )|, 2
2
2
(3.12)
where Cρ,α was defined in (3.6). Then we have
2 ∥∂ξ {χl m1 (t , ·)}∥L2 (R) ≤ Ct ξ e−Cρ,α (1+t )ξ χl 2
2 + C ξ e−Cρ,α (1+t )ξ χl 2
L (R)
−3/4
≤ Ct (1 + t )
1/4
≤ C (1 + t )
L (R)
−3/4
+ C (1 + t )
2 + C ξ e−Cρ,α (1+t )ξ χl′ 2
L (R)
−δ(1+t )
+ Ce
,
(3.13)
where we used Lemma 2.3 and the fact that
∥e−Cρ,α (1+t )ξ χl′ ∥L2 (R) ≤ ∥e−Cρ,α (1+t )ξ ∥L2 (ρ/2≤|ξ |≤ρ) 2
2
≤ e−Cρ,α (1+t ) −δ(1+t )
≤ Ce
ρ2 4
∥1∥L2 (ρ/2≤|ξ |≤ρ)
.
Since for ν(ξ ) = ξ + αξ − 1/4 4
2
√ √ √ (t −ν)cosh(t −ν) − sinh(t −ν) t , √ e− 2 (2ξ 3 + αξ )t 3 3 √(t −ν) √ √ ∂ξ m1 (t , ξ ) = (t ν) cos(t ν) − sin(t ν) t e− 2 (2ξ 3 + αξ )t 3 , √ (t ν)3
|ξ | < ζ (α), |ξ | > ζ (α),
and lim
y coshy − sinhy
y→0
y3
= lim
y cos y − sin y
y→0
y3
1
=− , 3
we have
∥∂ξ m1 (t , ·)∥L∞ (ρ/2≤|ξ |≤2R) ≤ Ce−δt . Then we obtain
∥∂ξ {χm m1 (t , ·)}∥L2 (R) ≤ Ce−δt .
(3.14)
For the high frequency part, it follows that
2ξ 3 + αξ |∂ξ {χh m1 (t , ξ )}| ≤ te 4 χ (ξ ) h ξ + αξ 2 − 41 3 2 ξ + αξ 1 t t −2 ′ −2 + e χh (ξ ) + e χh (ξ ) 3 1 ξ 4 + αξ 2 − 1 2 ξ 4 + αξ 2 − 4 4 ξ3 ξ3 t − 2t −δ t ≤ Ce 4 χ (ξ ) + Ce χh (ξ ) + Ce− 2 |χh′ (ξ )|. 3 ξ + αξ 2 − 14 h ξ 4 + αξ 2 − 1 2 4 − 2t
Note that
ξ3 ξ 4 + αξ 2 −
1 4
∈ L2 (R, ∞),
ξ3
2 3 ∈ L (R, ∞) ξ 4 + αξ 2 − 41 2
and suppχh′ (ξ ) ⊂ [R, 2R]. Therefore, we deduce that
∥∂ξ {χh m1 (t )}∥L2 (R) ≤ Ce−δt .
(3.15)
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Combining the estimates (3.13)–(3.15) yields
∥∂ξ m1 (t , ·)∥L2 (R) ≤ ∥∂ξ {χl m1 (t , ·)}∥L2 (R) + ∥∂ξ {χm m1 (t , ·)}∥L2 (R) + ∥∂ξ {χh m1 (t , ·)}∥L2 (R) 1
≤ C (1 + t ) 4 .
(3.16)
It follows from Lemma 2.1 with (3.11) and (3.16) that for q ∈ [1, ∞] 1
1
Mq (m1 (t , ·)) ≤ C ∥m1 (t , ·)∥L22 (R) ∥∂ξ m1 (t , ·)∥L22 (R) 1
1
≤ C (1 + t ) 8 (1 + t )− 8 = C , where C is independent of t. Therefore, we are led to (3.4):
∥K1 (t )g ∥Lq (R) = ∥F −1 [m1 g ]∥Lq (R) ≤ C ∥g ∥Lq (R) for any q ∈ [1, ∞]. This completes the proof of (3.2). For the proof of (3.1), recalling that
K0 (t )g = F −1 e
− 2t
e
it
−it ξ 4 +αξ 2 − 41
ξ 4 +αξ 2 − 14
+e 2(1 + ξ 2 )
(1 + ξ 2 ) g
and employing the Fourier multiplier m0 ( t , ξ ) = e
− 2t
e
it
−it ξ 4 +αξ 2 − 14
ξ 4 +αξ 2 − 14
+e 2(1 + ξ 2 )
,
we can prove easily from a modification similar to the above for m0 instead of m1 , then we omit the proof.
To deal with the nonlinear term of (1.1) including derivative, we use the following estimates with the second order derivative. We use the stationary phase method by Van der Corput to estimate the high frequency part. Proposition 3.2 (Lq –Lr Estimates With Second Order Derivative). For 2 ≤ q ≤ ∞, 1 ≤ r ≤ q ≤ ∞ such that t > 0, it follows that − 21
− 12
∥∂x2 K0 (t )g ∥Lq (R) ≤ C (1 + t ) ∥∂x2 K1 (t )g ∥Lq (R) ≤ C (1 + t )
1 1 r −q
−1
1 1 r − q −1
t
∥g ∥W 2,r (R) + Ce− 2 t t
∥g ∥Lr (R) + Ce− 2 t
− 12 1− 2q
− 12 1− 2q
∥g ∥W 2,q′ (R) ,
∥g ∥Lq′ (R) .
1 q
+
1 q′
= 1 and
(3.17) (3.18)
Proof. We decompose the linearized solution to three parts, K1 (t )g = F −1 [(χl + χm + χh )m1 (t , ·) g]
=: K1l (t )g + K1m (t )g + K1h (t )g , where K1j (t )g = F −1 [χj m1 (t , ·) g ] for j = l, m, h. We will show
∥∂
2 x K1l
(t )g ∥Lq (R) ≤ C (1 + t )
2 x K1m
∥∂
− 21
(t )g ∥Lq (R) ≤ Ce
−δ t t
1 1 r −q
−1
∥g ∥Lr (R) ,
(3.19)
∥g ∥Lr (R) ,
∥∂x2 K1h (t )g ∥Lq (R) ≤ Ce− 2 t
− 21 1− 2q
(3.20)
∥g ∥Lq′ (R) .
(3.21)
The estimate (3.19) is derived by the same way as the proof of Proposition 3.1. From (3.7) and Lemma 2.3
2 ∥∂x2 K1l (t )g ∥L∞ (R) ≤ C ξ 2 e−Cρ,α (1+t )ξ 1
L (|ξ |≤ρ)
∥g ∥L1 (R)
3
≤ C (1 + t )− 2 ∥g ∥L1 (R) . q
(3.22)
q
Next, we prove the L –L estimate of (3.19). It follows from the same computation as the above that
∥ξ 2 m1 (t , ·)χl ∥L2 (R) ≤ C
∞
e−2Cρ,α (1+t )ξ ξ 4 dξ 2
−∞ 5
≤ C (1 + t )− 4 ,
21
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and from (3.7) and (3.12) that
∥∂ξ {ξ 2 m1 (t , ·)χl }∥L2 (R) ≤ 2∥ξ m1 (t , ·)χl ∥L2 (R) + ∥ξ 2 ∂ξ {m1 (t , ·)χl }∥L2 (R) ≤ C ∥ξ e−Cρ,α (1+t )ξ ∥L2 (R) 2 2 2 + C t ∥ξ 3 e−Cρ,α (1+t )ξ ∥L2 (R) + ∥ξ 3 e−Cρ,α (1+t )ξ ∥L2 (R) + ∥ξ 2 e−Cρ,α (1+t )ξ χl′ ∥L2 (R) 2
3
7
7
≤ C (1 + t )− 4 + C {t (1 + t )− 4 + (1 + t )− 4 + e−δt } 3
≤ C (1 + t )− 4 . Then by Lemma 2.1 we deduce that 1/2
1/2
Mq (ξ 2 m1 (t , ·)χl ) ≤ C ∥ξ 2 m1 (t , ·)χl ∥L2 (R) ∥∂ξ {ξ 2 m1 (t , ·)χl }∥L2 (R) 5
3
≤ C (1 + t )− 8 − 8 = C (1 + t )−1 . Therefore, we arrive at
∥∂x2 K1l (t )g ∥Lq (R) ≤ C (1 + t )−1 ∥g ∥Lq (R)
(3.23)
for any q ∈ [1, ∞]. The interpolation between (3.22) and (3.23) yields (3.19). Next, we prove (3.20). Since (ρ/2) ≤ |ξ |2 ≤ (2R)2 , we have 2
∥ξ 2 m1 (t , ·)χm ∥L1 (R) ≤ C ∥m1 (t , ·)χm ∥L1 (R) ≤ Ce−δt , ∥ξ 2 m1 (t , ·)χm ∥L2 (R) ≤ C ∥m1 (t , ·)χm ∥L2 (R) ≤ Ce−δt , ∥∂ξ {ξ 2 m1 (t , ·)χm }∥L2 (R) ≤ C (∥m1 (t , ·)χm ∥L2 (R) + ∥∂ξ {m1 (t , ·)χm }∥L2 (R) ) ≤ Ce−δt , by virtue of (3.8), (3.10) and (3.14), and then
∥∂x2 K1m (t )g ∥L∞ (R) ≤ Ce−δt ∥g ∥L1 (R) ,
∥∂x2 K1m (t )g ∥Lq (R) ≤ Ce−δt ∥g ∥Lq (R) ,
which give the inequality (3.20). Finally we prove the estimate (3.21). By the Plancherel theorem, we see
2 −t ξ χ h 2 g e =C 1 4 2 ξ + αξ − 4 2
∥∂x2 K1h (t )g ∥L2 (R)
t
≤ Ce− 2 ∥g ∥L2 (R) .
L (R)
Thus the proof of (3.21) is reduced to verifying the inequality 1
t
∥∂x2 K1h (t )g ∥L∞ (R) ≤ Ce− 2 t − 2 ∥g ∥L1 (R) . To show (3.24), we first decompose ∂
2 x K1h
e
it
(t )g into two parts,
ξ 4 +αξ 2 − 14
t ∂x2 K1h (t )g = −e− 2 F −1
= −e
−it ξ 4 +αξ 2 − 14
−e
2i ξ 4 + αξ 2 − it
χh ξ 2
1 4
e
ξ 4 +αξ 2 − 41
t
t
= −e− 2 J+ (t , ·) ∗ g + e− 2 J− (t , ·) ∗ g , where we set ±it ξ 4 +αξ 2 − 41
J± (t , x) = F −1
e
2i ξ 4 + αξ 2 −
1
= (2π )− 2
g
−it ξ 4 +αξ 2 − 14 2 2 χ ξ e χ ξ t h h F −1 g + e − 2 F −1 g 2i ξ 4 + αξ 2 − 41 2i ξ 4 + αξ 2 − 41
− 2t
(3.24)
e R
ixξ ±it
1 4
ξ 2 χh
ξ 4 +αξ 2 − 14
ξ 2 χh (ξ ) dξ . 2i ξ 4 + αξ 2 − 41
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To apply Lemma 2.2 to J± as
φ(ξ ) =
xξ t
1
ξ 4 + αξ 2 − ,
±
ψ(ξ ) =
4
ξ 2 χh (ξ )
2i ξ 4 + αξ 2 −
1 4
,
we check that φ and ψ satisfy the assumptions of Lemma 2.2. We immediately have ψ ∈ L∞ (R) and
∂ξ ψ(ξ ) =
(2αξ 3 − ξ )χh (ξ ) ξ 2 χh′ (ξ ) + 3 / 2 4i ξ 4 + αξ 2 − 14 2i ξ 4 + αξ 2 −
∈ L1 (R, ∞).
1 4
Further, the direct computation yields
′′ |φ (ξ )| = ∂ξ2 ξ 4 + αξ 2 −
2ξ 6 + 3αξ 4 − 3ξ 2 /2 − α/4 . = 3 4 ξ 4 + αξ 2 − 1 2 1
4
Choosing sufficiently large R ≫ 1 such that 3αξ − 3ξ /2 − α/4 > 0 and ξ 4 + αξ 2 − 1/4 < 3ξ 4 /2 for |ξ | ≥ R, we have 4
2ξ 6
|φ ′′ (ξ )| > 3
2
ξ4
2
2
23 = 3 32 > 1. 2
Then, applying Lemma 2.2 for arbitrary M ≫ R, we obtain
M
e (
it xt ξ ±φ(ξ )
R
M ′ ) ψ(ξ )dξ ≤ 10|t |−1/2 |ψ(M )| + |ψ (ξ )| d ξ R ∞ −1/2 ′ ≤ C |t | ∥ψ∥L∞ (R) + |ψ (ξ )|dξ R
≤ C |t |−1/2 . Since the above C is independent of M, tending M → ∞ yields
∞
R
eit ( t ξ ±φ(ξ )) ψ(ξ )dξ ≤ C |t |− 2 . x
1
In the same fashion, we also have
−R
e (
it xt ξ ±φ(ξ )
−∞
) ψ(ξ )dξ ≤ C |t |− 21 .
Combining these, we obtain 1
|J± (t , x)| ≤ C |t |− 2 . Therefore, it follows from the Hausdorff–Young inequality that 1
∥J± (t ) ∗ g ∥L∞ (R) ≤ C ∥J± (t )∥L∞ (R) ∥g ∥L1 (R) ≤ C |t |− 2 ∥g ∥L1 (R) , which completes the proof of (3.24). Consequently, the desired estimate (3.18) is obtained. We omit the proof of (3.17), since the proof is similar to the above. To deal with the nonlinear part without regularity loss we shall need the following estimate. The proof is essentially the same as that of Proposition 3.1. Then we omit it. Corollary 3.3. For 1 ≤ r ≤ q ≤ ∞, 2 ≤ q ≤ ∞ and t > 0, it follows that − 12
∥∂x3 K1 (t )g ∥Lq (R) ≤ C (1 + t )
1 1 r −q
− 23
t
∥g ∥Lr (R) + Ce− 2 t
− 12 1− 1q
∥∂x g ∥Lq′ (R) .
4. Global existence In this section, we show the global existence of solution for (1.12). Our proof is based on the contraction mapping principle with the help of the linear estimates introduced in Section 3.
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Proof of Theorem 1.1. Let us introduce the mapping
Ψ [u](t ) = K (g0 , g1 ) +
t
K1 (t − s)∂x f (∂x u(s))ds, 0
for K (g0 , g1 ) = K0 (t )g0 + K1 (t )
1 2
g0 + g1
,
and the complete metric space
1 2 X = u(t ) ∈ C [0, ∞); L (R) ∩ H (R) ; ∥u∥X ≤ N , with
∥u∥X = sup
t ∈[0,∞)
5 ∥u(t )∥L1 (R) + (1 + t ) 4 ∥∂x2 u(t )∥L2 (R) ,
where N is determined later. We shall prove the mapping Ψ is contraction in X . Thus we concentrate on showing the following claims: (I) ∥Ψ [u]∥X ≤ N for u ∈ X . (II) ∥Ψ [u] − Ψ [v]∥X ≤ 12 ∥u − v∥X for u, v ∈ X . Setting
δ0 = ∥g0 ∥W 2,1 (R)∩H 2 (R) + ∥g1 ∥L1 (R)∩L2 (R) , by (3.1) and (3.2), we have
∥K (g0 , g1 )∥L1 (R) ≤ C ∥g0 ∥W 2,1 (R) + ∥g1 ∥L1 (R) ≤ C δ0 .
(4.1)
It follows from the Gagliardo–Nirenberg inequality (see e.g. [3, Theorem 1.3.7]) that
∥∂x u∥L2(p−1) (R) ≤ C ∥u∥ 1
1 1 1+ p− 5 1 L1 (R)
1 4 4− p− 5 1 2 x u L2 (R)
,
∥∂ ∥
(4.2)
4
∥∂x u∥L∞ (R) ≤ C ∥u∥L51 (R) ∥∂x2 u∥L52 (R) .
(4.3)
Applying (4.2), we obtain
∥∂x f (∂x u(s))∥L1 (R) ≤ C f ′ (∂x u(s))∂x2 u(s)L1 (R) ≤ C ∥∂x u(s)∥pL2−(p1−1) (R) ∂x2 u(s)L2 (R) p 4p ≤ C ∥u(s)∥ 51 ∂ 2 u(s) 52 L (R)
L (R)
x
p 5 L1 (R)
≤ C (1 + s)−p ∥u(s)∥
4p5 5 (1 + s) 4 ∂x2 u(s)L2 (R)
≤ C (1 + s)−p ∥u∥pX , which gives t
0
∥K1 (t − s)∂x f (∂x u(s))∥L1 (R) ds ≤ C
t
∥∂x f (∂x u(s))∥L1 (R) ds t ≤ C ∥u∥pX (1 + s)−p ds 0
0 p X
≤ C ∥ u∥ .
(4.4)
Combining the estimates (4.1) and (4.4), we have
∥Ψ [u](t )∥L1 (R) ≤ ∥K (g0 , g1 )∥L1 (R) + ≤ C δ0 + C ∥u∥pX .
t
0
∥K1 (t − s)∂x f (∂x u(s))∥L1 (R) ds (4.5)
H. Takeda, S. Yoshikawa / J. Math. Anal. Appl. (
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From (4.3) we obtain
−1 2 ∥∂x f (∂x u(s))∥L2 (R) ≤ C ∥∂x u(s)∥pL∞ (R) ∂x u(s) L2 (R) p−1 4(p−1) +1 ≤ C ∥u(s)∥ 15 ∂ 2 u(s) 2 5 L (R)
L (R)
x
1
≤ C (1 + s)−p− 4 ∥u(s)∥
p 5 L1 (R)
4p5 + 15 5 (1 + s) 4 ∂x2 u(s)L2 (R)
1
≤ C (1 + s)−p− 4 ∥u∥pX .
(4.6)
It then follows from (3.1) and (3.2) that
2 ∂ K (g0 , g1 ) 2
L (R)
x
t 5 ≤ C (1 + t )− 4 ∥g0 ∥W 2,1 (R) + ∥g1 ∥L1 (R) + Ce− 2 ∥g0 ∥H 2 (R) + ∥g1 ∥L2 (R) 5
≤ C (1 + t )− 4 δ0 and that t
0
t
t t −s 5 (1 + t − s)− 4 ∥∂x f (∂x u(s))∥L1 (R) ds + C e− 2 ∥∂x f (∂x u(s))∥L2 (R) ds 0 0 t t t −s 1 p p − 54 −p ≤ C ∥ u∥ X (1 + t − s) (1 + s) ds + C ∥u∥X e− 2 (1 + s)−p− 4 ds 0 0 5 1 ≤ C (1 + t )− 4 + (1 + t )−p− 4 ∥u∥pX
2 ∂ K1 (t − s)∂x f (∂x u(s)) 2 ds ≤ C x L (R)
5
≤ C (1 + t )− 4 ∥u∥pX , by virtue of Lemma 2.4. Therefore, we have
∥∂x2 Ψ [u](t )∥L2 (R) ≤ ∂x2 K (g0 , g1 )L2 (R) + 5
t
2 ∂ K1 (t − s)∂x f (∂x u(s)) 2
L (R)
x
0 5
≤ C (1 + t )− 4 δ0 + C (1 + t )− 4 ∥u∥pX .
ds (4.7)
Consequently, the estimates (4.5) and (4.7) show for constants C0 and C
∥Ψ [u]∥X ≤ C0 δ0 + C ∥u∥pX ≤ C0 δ0 + CN p . In a similar fashion, it can be shown that
∥Ψ [u] − Ψ [v]∥X ≤ C (∥u∥pX−1 + ∥v∥pX−1 )∥u − v∥X ≤ 2CN p−1 ∥u − v∥X . Let us define N by N = 2C0 δ0 . Choosing δ0 sufficiently small satisfying 4CN p−1 < 1, we obtain the claim (I) and (II). Thereby, from the Banach fixed point theorem, there exists a unique fixed point u ∈ X such that u(t ) = Ψ [u](t ), which completes the proof.
In Theorem 1.1 the persistence property of solution does not hold because u ̸∈ C ([0, ∞); W 2,1 (R) ∩ H 2 (R)). However, in an argument similar to the proof of Theorem 1.1 we can also obtain the global well-posedness result for H 2 -solution. Corollary 4.1 (Global Well-Posedness). Assume that the nonlinear term f satisfies the assumptions (1.3). If the initial data (g0 , g1 ) ∈ H 2 (R) × L2 (R) and
∥g0 ∥H 2 (R) + ∥g1 ∥L2 (R) is sufficiently small, then there exists a unique mild solution u(t ) for the initial value problem (1.1)–(1.2) in the class C ([0, ∞); H 2 (R)) satisfying
∥u(t )∥L2 (R) ≤ C ,
∥∂x2 u(t )∥L2 (R) ≤ C (1 + t )−1 ,
t ≥ 0.
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)
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5. Smoothing of the solution In the previous section, we construct the time-global solution for the integral equation (1.12). In this section, we investigate the smoothing effect of the solution. Proof of Theorem 1.3. For 2 ≤ q ≤ ∞, we have
∥∂x2 u(t )∥Lq (R) ≤ ∥∂x2 K (g0 , g1 )∥Lq (R) +
t
∥∂x2 K1 (t − s)∂x f (∂x u(s))∥Lq (R) ds. 0
From Proposition 3.2 we have − 21 1− 1q −1
∥∂x2 K (g0 , g1 )∥Lq (R) ≤ C (1 + t )
≤ Ct
− 12 1− 1q
−1
t
∥(g0 , g1 )∥W 2,1 ×L1 (R) + e− 2 t
− 12 1− 2q
∥(g0 , g1 )∥W 2,1 (R)∩H 2 (R)×L1 (R)∩L2 (R)
∥(g0 , g1 )∥W 2,1 (R)∩H 2 (R)×L1 (R)∩L2 (R) .
We shall show that the nonlinear part decays more rapid than the linear part. From Proposition 3.2 and Corollary 3.3 we have t
∥∂
2 x K1
(t − s)∂x f (∂x u(s))∥
t 2
Lq (R)
∥∂x3 K1 (t − s)f (∂x u(s))∥Lq (R) ds
ds ≤ 0
0
t
∥∂x2 K1 (t − s)∂x f (∂x u(s))∥Lq (R) ds
+ t 2 t 2
− 21 1− 1q − 32
(1 + t − s)
≤C 0
t
e−
+C
t −s 2
∥f (∂x u(s))∥L1 (R) ds
− 12 1− 2q
( t − s)
0 t
+C t 2
(1 + t − s)
− 12
1 − 1q 2
∥∂x f (∂x u(s))∥Lq′ (R) ds
−1
∥∂x f (∂x u(s))∥L2 (R) ds.
(5.1)
It follows from the Gagliardo–Nirenberg inequality (see e.g. [3, Theorem 1.3.7]) that for any r ∈ [4/3, ∞]
1− 25 2− 1r
∥∂x u(s)∥Lr (R) ≤ C ∥u(s)∥L1 (R) ≤ C (1 + s)
2 2− 1r 5
∥∂x2 u(s)∥L2 (R)
− 21 1− 1r − 12
,
(5.2)
with the help of (1.15). Then, we obtain t 2
− 21 1− 1q − 32
(1 + t − s)
0
∥f (∂x u(s))∥L1 (R) ds ≤ C 1 +
t
− 12
1− 1q − 32
t 2
2
0
≤ C (1 + t )
− 12 1− 1q
− 32
≤ C (1 + t )
− 12 1− 1q
− 32
t 2
∥∂x u(s)∥pLp (R) ds 1
(1 + s)−p+ 2 ds
0 t 2
3
(1 + s)− 2 ds
0
≤ C (1 + t )
− 12 1− 1q − 32
,
where C is independent of t. Next, we calculate the second term in the right hand side of (5.1). Since 1 ≤ q′ ≤ 2, we have t
e
s − t− 2
− 21 1− 2q
(t − s)
0 t
e−
≤C
t −s 2
∥∂x f (∂x u(s))∥Lq′ (R) ds ≤ C
− 21 1− 2q
(t − s)
1
(1 + s)−p− 2q ds
0 1
≤ C (1 + t )−p− 2q ≤ C (1 + t )
− 12 1− 1q − 32
,
t
s − t− 2
e 0
( t − s)
− 12 1− 2q
∥∂x u(s)∥p−2q1 L q−2
(p−1)
(R)
∥∂x2 u(s)∥L2 (R) ds
H. Takeda, S. Yoshikawa / J. Math. Anal. Appl. (
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by virtue of the Holder inequality, the estimate (5.2) and Lemma 2.4(ii). Lastly, the third term of the right hand side of (5.1) can be estimated by t
t 2
(1 + t − s)
− 21
1 − 1q 2
−1
∥∂x f (∂x u(s))∥L2 (R) ds ≤ C
t
t 2
(1 + t − s)
− 12
1 − 1q 2
−1
1
(1 + s)−p− 4 ds
1
≤ C (1 + t )−p− 4 log(2 + t ) − 12 1− 1q − 32
≤ C (1 + t )
− 12 1− 1q − 32
≤ C (1 + t ) due to (4.6). This completes the proof of Theorem 1.3.
1
1
{(1 + t )− 2q − 4 log(2 + t )} ,
Acknowledgments The authors should express deep gratitude to Professors Takayoshi Ogawa, Gustavo Ponce and Yoshio Tsutsumi for their useful advice. The authors are grateful to Professor Hideo Kozono for his many helpful suggestions. The authors are partially supported by the Japan Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Young Scientists (B), Nos 22740097 and 23740116. References [1] P. Brenner, V. Thomée, L. Wahlbin, Besov Spaces and Applications to Difference Methods for Initial Value Problems, in: Lecture Notes in Mathematics, vol. 434, Springer-Verlag, Berlin, New York, 1975. [2] M. Brokate, J. Sprekels, Hysteresis and Phase Transitions, in: Applied Mathematical Sciences, vol. 121, Springer-Verlag, New York, 1996. [3] T. Cazenave, Semilinear Schrödinger Equations, in: Courant Lecture Notes in Mathematics, vol. 10, American Mathematical Society, Providence, RI, 2003, New York University, Courant Institute of Mathematical Sciences, New York. [4] J.D. Evans, V.A. Galaktionov, J.F. Williams, Blow-up and global asymptotics of the limit unstable Cahn–Hilliard equation, SIAM J. Math. Anal. 38 (2006) 64–102. [5] M. Grasselli, G. Schimperna, A. Segatti, S. Zelik, On the 3D Cahn–Hilliard equation with inertial term, J. Evol. Equ. 9 (2009) 371–404. [6] M. Grasselli, G. Schimperna, S. Zelik, On the 2D Cahn–Hilliard equation with inertial term, Comm. Partial Differntial Equations 34 (2008) 137–170. [7] N. Hayashi, E. Kaikina, P.I. Naumkin, On the critical nonlinear damped wave equation with large initial data, J. Math. Anal. Appl. 334 (2007) 1400–1425. [8] N. Hayashi, E. Kaikina, P.I. Naumkin, Damped wave equation with super critical nonlinearities, Differential Integral Equations 17 (2004) 637–652. [9] T. Hosono, T. Ogawa, Large time behavior and Lp –Lq estimate of 2-dimensional nonlinear damped wave equations, J. Differential Equations 203 (2004) 82–118. [10] G. Karch, Selfsimilar profiles in large time asymptotics of solutions to damped wave equations, Studia Math. 143 (2000) 175–197. [11] S. Kawashima, M. Nakao, K. Ono, On the decay property of solutions to the Cauchy problem of the semilinear wave equation with a dissipative term, J. Math. Soc. Japan 47 (1995) 617–653. [12] C.E. Kenig, G. Ponce, L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J. 40 (1991) 33–69. [13] T.T. Li, Y. Zhou, Breakdown of solutions to u + ut = |u|1+α , Discrete Contin. Dyn. Syst. 1 (1995) 503–520. [14] Z. Liu, S. Zheng, Semigroups associated with dissipative systems, in: Chapman & Hall/CRC Research Notes in Mathematics, vol. 398, Chapman & Hall/CRC, Boca Raton, FL, 1999. [15] P. Marcati, K. Nishihara, The Lp –Lq estimates of solutions to one-dimensional damped wave equations and their application to compressible flow through porous media, J. Differential Equations 191 (2003) 445–469. [16] T. Narazaki, Lp –Lq estimates for damped wave equations and their applications to semi-linear problem, J. Math. Soc. Japan 56 (2004) 585–626. [17] K. Nishihara, Lp –Lq estimates for the 3-D damped wave equation and their application to the semilinear problem, Sem. Notes Math. Sci. Ibaraki Univ. 6 (2003) 69–83. [18] K. Nishihara, Lp –Lq estimates of solutions to the damped wave equation in 3-dimensional space and their application, Math. Z. 244 (2003) 631–649. [19] K. Nishihara, H. Zhao, Decay properties of solutions to the Cauchy problem for the damped wave equation with absorption, J. Math. Anal. Appl. 313 (2006) 598–610. [20] T. Ogawa, H. Takeda, Large time behavior of solutions for a system of nonlinear damped wave equations, J. Differential Equations 251 (2011) 3090–3113. [21] R. Racke, C. Shang, Global attractors for nonlinear beam equations. Preprint. [22] I. Segal, Dispersion for non-linear relativistic equations, II, Ann. Sci. E’cole Norm. Sup. 1 (1968) 459–497. [23] H. Takeda, Global existence of solutions for higher order nonlinear damped wave equations, in: Dynamical Systems, Differential Equations and Applications. 8th AIMS Conference, suppl., in: Discrete Contin. Dyn. Syst., 2011, pp. 1358–1367. [24] H. Takeda, S. Yoshikawa, Asymptotic profile of solutions for the limit unstable Cahn–Hilliard equation with inertial term, Differential Integral Equations 25 (2012) 341–362. [25] H. Takeda, S. Yoshikawa, On the initial value problem of the semilinear beam equation with weak damping II: asymptotic profiles, J. Differential Equations 253 (2012) 3061–3080. [26] H. Takeda, S. Yoshikawa, On the decay property of solutions to the Cauchy problem of the semilinear beam equation with weak damping for large initial data, Adv. Stud. Pure Math. (in press). [27] T. Yamada, Improvement of convergence rates for a parabolic system of chemotaxis in the whole space, Math. Methods Appl. Sci. 34 (2011) 2103–2124. [28] Q.S. Zhang, A blow-up result for a nonlinear wave equation with damping: the critical case, C. R. Acad. Sci. Paris Ser. I Math. 333 (2001) 109–114.
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Journal of Mathematical Analysis and Applications journal homepage: www.elsevier.com/locate/jmaa
On the initial value problem of the semilinear beam equation with weak damping I: Smoothing effect Hiroshi Takeda a,∗ , Shuji Yoshikawa b a
Department of Intelligent Mechanical Engineering, Faculty of Engineering, Fukuoka Institute of Technology, 3-30-1 Wajiro-higashi, Higashi-ku, Fukuoka, 811-0295, Japan b
Department of Engineering for Production and Environment, Graduate School of Science and Engineering, Ehime University, 3 Bunkyo-cho, Matsuyama, Ehime, 790-8577, Japan
article
abstract
info
Article history: Received 15 March 2012 Available online xxxx Submitted by David Russell Keywords: Damped beam equation Cauchy problem Fourth order wave equation with damping Asymptotic profile Perturbation
This paper is devoted to the study of the initial value problem for some semilinear damped beam equation. We shall prove the unique global existence of a small solution for the equation and also obtain a certain smoothing effect of the solution. Our method is based on the expansion formula for the propagator to decompose into the dissipative part and the dispersive part. To control the dispersive part without regularity loss, we apply the stationary phase method developed by Van der Corput. As a result, we observe the smoothing effect of the solution which does not occur on the solution for damped wave equations. © 2012 Elsevier Inc. All rights reserved.
1. Introduction In this paper we study the initial value problem for the following semilinear damped beam equation
∂t2 u + ∂t u + ∂x4 u − α∂x2 u = ∂x f (∂x u), u(0, x) = g0 (x),
∂t u(0, x) = g1 (x),
t > 0, x ∈ R, x ∈ R,
(1.1) (1.2)
where u = u(t , x); (0, T )× R → R is an unknown function, g0 (x) and g1 (x) are given initial data and α is a positive constant. We assume that the nonlinear function f ∈ C 1 (R) satisfies that for some p ≥ 2 and small any v , v1 and v2 ∈ R,
|f (v)| ≤ C |v|p ,
|f ′ (v)| ≤ C |v|p−1 , |f (v1 ) − f (v2 )| ≤ C |v1 |p−1 + |v2 |p−1 |v1 − v2 |, |f ′ (v1 ) − f ′ (v2 )| ≤ C |v1 |p−2 + |v2 |p−2 |v1 − v2 |,
(1.3)
where C is independent of v , v1 and v2 . The typical example is given by f (v) = −|v|p−1 v,
p ≥ 2.
The lowest exponent 2 for p is needed for all the powers in (1.3) to be non-negative.
∗
Corresponding author. E-mail addresses: [email protected] (H. Takeda), [email protected] (S. Yoshikawa).
0022-247X/$ – see front matter © 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2012.12.015
(1.4)
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Our aim of this paper is to show the existence of a global solution u ∈ C ([0, ∞); L1 (R) ∩ H 2 (R)) of (1.1)–(1.2) for small data (g0 , g1 ) ∈ W 2,1 (R) ∩ H 2 (R) × L1 (R) ∩ L2 (R) that has the decay rates
∥u(t )∥
Lq (R)
− 21 1− 1q
≤ C (1 + t )
∥∂x2 u(t )∥Lq (R) ≤ Ct
− 12 1− 1q
−1
,
t ≥ 0, 1 ≤ q ≤ ∞,
,
t > 0, 2 ≤ q ≤ ∞.
(1.5) (1.6)
According to the result in [10,23], the solution of the initial value problem for the linear equation
∂t2 u + ∂t u + (−∂x2 )ℓ u = 0,
t > 0, x ∈ R, ℓ ∈ N
decays − 21ℓ 1− 1q
∥u(t )∥Lq (R) ≤ C (1 + t )
,
t ≥ 0, 1 ≤ q ≤ ∞.
Then the decay estimate (1.5) coincides with that of the damped wave equation
∂t2 u + ∂t u − ∂x2 u = 0,
t > 0, x ∈ R,
(1.7)
which means that the decay property of the solution to (1.1)–(1.2) is mainly dominated by the lower order term α∂x2 u (α > 0). For the corresponding nonlinear problem to (1.1)–(1.2) in α = 0, we refer to [23,24]. The decay estimate (1.6) implies that the solution also belongs to the class C ((0, ∞); W 2,∞ (R)). Such a smoothing effect arises from the dispersive property and never occurs on the solution for damped wave equations under the above data. Moreover, in the sequel [25], we shall give several asymptotic profiles of the decaying solution. As a consequence, we see that the decay rate (1.5) and (1.6) are optimal. Before discussing the result of this paper more precisely, we shall explain the background of our problem and the related results. The linear beam equation with damping
∂t2 u + (−∂x2 )m ∂t u + ∂x4 u = 0 is studied by many authors (see [14] and reference therein). In particular, the equation is called weak damped type when m = 0. Under the constant distributed temperature T higher than the critical temperature Tc of martensitic phase transitions in shape memory alloys, the Falk model for the displacement u is written as
∂t2 u + ∂x4 u − α∂x2 u = ∂x f (∂x u), where α = a1 (T − Tc ) and f (v) = −a2 |v|p2 −1 v + a3 |v|p3 −1 v
(1.8)
for positive physical constants a1 , a2 and a3 with p2 = 3 and p3 = 5. We refer to the monograph [2, Chapter 5] for more precise information and the derivation of the Falk model. The initial boundary value problem of the equation with weak damping (i.e. (1.1) with (1.8)) was studied in [21]. They obtained existence of time global solution with large data and its global attractor in bounded spatial domain. It plays an essential role in their proof that F (v) = a3 |v|p3 +1 /(p3 + 1) − a2 |v|p2 +1 /(p2 + 1) is bounded from below and the boundedness of the spatial domain. The double-well form (1.8) also appears in the Cahn–Hilliard equation
∂t u + ∆2 u = ∆f (u).
(1.9)
Evans–Galaktionov–Williams [4] studied the Cahn–Hilliard equation (1.9) of limit case as a3 → 0 called limit unstable case. On the other hand, they called the case a2 → 0 limit stable case. To sum up, for f (v) = a|v|p−1 v , a > 0 and a < 0 are regarded as the limit stable case and the limit unstable case, respectively. Grasselli et al. in [6,5] considered the Cahn–Hilliard equation with inertial term
∂t2 u + ∂t u + ∆2 u = −∆f (u),
t > 0, x ∈ Ω (b Rn , n = 2, 3).
In [24] authors studied the Cauchy problem of limit unstable Cahn–Hilliard equation with inertial term
∂t2 u + ∂t u + ∆2 u = −∆(|u|p−2 u),
t > 0 , x ∈ Rn ,
(1.10)
for n = 1, 2, 3. As mentioned above, the solution for (1.10) decays − 4n 1− 1q
∥u(t )∥Lq (Rn ) ≤ C (1 + t )
,
t ≥ 0, 2 ≤ q ≤ ∞.
We can observe that the rate of the decay estimate is slower than (1.5), due to the absence of the second order term α ∆u. Moreover, we note that in this article we also obtain the smoothing estimate (1.6) restricting ourselves to the onedimensional case. The same strategy does not work in the multi-dimensional case. Correspondingly, for the semilinear damped wave equation
∂t2 u + ∂t u − ∆u = a|u|p−1 u,
t > 0, x ∈ Rn ,
(1.11)
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the right hand side is called the absorbing term when a < 0 and the sourcing term when a > 0. These terms play completely different roles and these analyses are also different. Indeed, making use of the energy method, the absorbing type (1.11) with a < 0 have a solution without the smallness assumption of the data, and further the decay estimate for the solution can be obtained (see e.g. [7,11,19]). On the other hand, in the case of the sourcing type with a > 0, for a certain small p the blow-up result was shown by Nishihara [17] (we also refer to [13,28]). It is also known that the small data global existence and the asymptotic profile of the solution to (1.11) for a certain large p is given by the heat kernel (see e.g. [8,9,20] and reference therein). The Eq. (1.1) with (1.4) corresponds to the limit unstable case of the Falk model with weak damping. From the mathematical point of view the limit stable case and limit unstable case of (1.1) correspond to the absorbing type and the sourcing type of (1.11), respectively. For (1.1) of the limit stable case, authors have shown that there exists time global solution with decay for large data in [26]. In this study we are interested in the Eq. (1.1) of the limit unstable case (i.e. (1.1) with (1.4)). For the problem the large data global existence is no longer expected. Let us state our main results of this paper. We call u(t ) a mild solution of the initial value problem (1.1)–(1.2) if u(t ) ∈ C ([0, ∞); L1 (R) ∩ H 2 (R)) and satisfies the integral equation u(t ) = K0 (t )g0 + K1 (t )
1 2
g0 + g1
t
+
K1 (t − s)∂x f (∂x u(s))ds,
(1.12)
0
where the evolution operators K0 (t ) and K1 (t ) are given by
K0 (t )f = F
−1
− 2t
e
it
ξ 4 +αξ 2 − 14
+e
−it ξ 4 +αξ 2 − 14
2
K1 (t )g = F −1 e
e
− 2t
e
it
ξ 4 +αξ 2 − 14
−e
−it ξ 4 +αξ 2 − 41
2i ξ 4 + αξ 2 −
1 4
f,
(1.13)
g .
(1.14)
The following result is the global existence of a mild solution for small data with the decay. Theorem 1.1 (Unique Global Existence of Solution). Assume that f satisfies the assumption (1.3) for p ≥ 2. If the initial data
(g0 , g1 ) ∈ W 2,1 (R) ∩ H 2 (R) × L1 (R) ∩ L2 (R) and ∥g0 ∥W 2,1 (R)∩H 2 (R) + ∥g1 ∥L1 (R)∩L2 (R)
is sufficiently small, then there exists a unique mild solution u(t ) for the initial value problem (1.1)–(1.2) in C ([0, ∞); L1 (R) ∩ H 2 (R)) satisfying
∥u(t )∥L1 (R) ≤ C ,
5
∥∂x2 u(t )∥L2 (R) ≤ C (1 + t )− 4 ,
t ≥ 0.
(1.15)
Remark 1.2. Theorem 1.1 states the existence of a kind of weak solutions for (1.1)–(1.2). On the other hand, we can prove the existence of a solution in C ([0, ∞); W s,1 (R) ∩ H s+2 (R)) for s ≥ 0 under the corresponding additional regularity assumption of the data and the nonlinearity f , by the same procedure as the proof of Theorem 1.1. Our next purpose is to show the smoothing effect of the global in time solution for (1.1)–(1.2) constructed in Theorem 1.1. Theorem 1.3 (Smoothing Effect). Under the same condition as in Theorem 1.1, the mild solution u(t ) to (1.1)–(1.2) satisfies u(t ) ∈ C ([0, ∞); L1 (R) ∩ H 2 (R)) ∩ C ((0, ∞); W 2,∞ (R)). Moreover, the following time decay estimate holds
∥∂x2 u(t )∥Lq (R) ≤ Ct
− 21 1− 1q −1
,
2 ≤ q ≤ ∞, t > 0.
(1.16)
In Theorem 1.3, ‘‘smoothing effect’’ means the time global decay of the solution in the sense of W 2,∞ for the initial data (g0 , g1 ) ∈ W 2,1 ∩ H 2 × L1 ∩ L2 . The smoothing arises from the dispersive property of the fourth order term, since the estimate (1.16) does not hold for the solution of the damped wave equation (1.7) under the same assumption in Theorem 1.3. Our strategy for the proof of theorems lies in the application of the Lq –Lr estimates for the linearized solution of the damped beam equation. To obtain the estimates, we use the Fourier splitting method and decompose the linearized solution into dissipative and dispersive parts.
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Nishihara in [18] (see also [9,15,16]) showed that for the solutions of linear heat and wave equations
∂t v − ∆v = 0, v(0, x) = g0 (x) + g1 (x),
t > 0 , x ∈ Rn , x ∈ Rn ,
2 ∂t w − ∆w = 0, w(0, x) = g0 (x), ∂t w(0, x) = g1 (x),
t > 0, x ∈ Rn , x ∈ Rn ,
the solution of the damped wave equation (1.7) can be decomposed into v(t ) and w(t ) such as t
u(t ) = v(t ) + e− 2 w(t ) + error
as t → ∞.
(1.17)
This expansion (1.17) states that the factor of w(t ) in the solution is asymptotically small when t is large. In contrast, we
established the decomposition of K1 (t ) and its second order derivative ∂x2 K1 (t ) into the dissipative evolution et α∂x and the 2
t
dispersive part e− 2 e±it ∂x such as 2
t
K1 (t ) = et α∂x + e− 2 ∂x−2 e±it ∂x + error 2
∂
2 x K1
2 t α∂x2 xe
(t ) = ∂
t
2
+e
− 2t ±it ∂x2
e
as t → ∞,
(1.18)
+ error as t → ∞,
where e− 2 e±it ∂x decays faster than the dissipative part. Indeed, the decay order of the solution is dominated by the dissipative part without ‘‘dispersive singularities’’ and we can obtain the decay estimates as the solution of the linear heat equation. On the other hand, we need to pay attention to the dispersive part, since the regularity property of the solution mainly depends on the high frequency part, in general. Then our analysis for the dispersive part, which is based on the stationary phase method, makes us possible to control the L∞ -norm of ∂x2 K1 (t ) without the additional regularity assumption for the data. Summing up the above, the expansion (1.18) enables us to obtain the smoothing property and the asymptotic profile for our global solution constructed in Theorem 1.1. We conclude the introduction by giving notation used in this paper. We denote by ∂y partial differential operator with respect to a variable y. Let W k,p (R) be the Sobolev space for k ∈ N and 1 ≤ p ≤ ∞ defined by 2
W k,p (R) = {u : R → R; ∥u∥W k,p (R) = ∥u∥Lp (R) + ∥∂xk u∥Lp (R) < ∞}, and H k (R) = W k,2 (R). We denote the Fourier and the Fourier inverse transforms by F and F −1 . We also let f the Fourier transform of f . We denote the several positive constants by C > 0 without confusions. These constants may change from line to line. 2. Preliminaries For f ∈ L2 (Rn ) ∩ Lq (Rn ), 1 ≤ q ≤ ∞, let m(ξ ) be the Fourier multiplier defined by n
F −1 [m f ](x) = (2π )− 2
Rn
eix·ξ m(ξ ) f (ξ )dξ .
We define Mq as the class of the Fourier multiplier for 1 ≤ q ≤ ∞:
Mq =
m : Rn → R, there exists a positive constant Cq < ∞ such that ∥F −1 [m f ]∥Lq (Rn ) ≤ Cq ∥f ∥Lq (Rn ) ,
and for m ∈ Mq , we let Mq (m) = sup f ̸=0
∥F −1 [m f ]∥Lq (Rn ) . q ∥f ∥L (Rn )
The following lemma is known as the Carleson–Beurling inequality, which is applied to show the Lq –Lq boundedness of the Fourier multiplier. Lemma 2.1 (The Carleson–Beurling Inequality [1]). If m ∈ H s (Rn ) with s > following estimate holds 1− n
n , 2
then m ∈ Mq for all 1 ≤ q ≤ ∞. Moreover, the
n
Mq (m) ≤ C ∥m∥L2 (R2sn ) ∥∂ξs m∥L2s2 (Rn ) . We introduce the following classical lemma by Van der Corput to control the oscillation integral without the additional regularity assumption.
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Lemma 2.2 ([12, Lemma 2.8]). Assume φ ∈ C 2 ([a, b]), |φ ′′ (ξ )| > 1 for ξ ∈ [a, b]. If |ψ(b)| < ∞ and ψ ′ ∈ L1 (a, b), then,
b
e
it φ(ξ )
a
− 12 |ψ(b)| + ψ(ξ )dξ ≤ 10|t |
b
|ψ (ξ )|dξ . ′
a
To compute the decay order of the linear estimates, we make an extensive use of the following well-known estimates. Lemma 2.3. For d ∈ N ∪ {0} and positive constant β , there exists a constant C depending only on d and β such that
1
|ξ |d e−β(1+t )|ξ | dξ ≤ C (1 + t )− 2 (1+d) . 2
R
The next lemma is useful to compute the decay order of nonlinear part in the integral equation (1.12). Lemma 2.4. Given any t ≥ 0, the following properties hold:
(i) Let a > 0 and b > 0 with min{a, b} > 1. There exists a constant C depending only on a and b such that t (1 + t − s)−a (1 + s)−b ds ≤ C (1 + t )− min{a,b} .
(2.1)
0
(ii) Let 1 > a ≥ 0, b > 0 and c > 0. There exists a constant C independent of t such that t e−c (t −s) (t − s)−a (1 + s)−b ds ≤ C (1 + t )−b .
(2.2)
0
For the proof of (2.1), we refer to Segal [22]. The estimate (2.2) is well-known (see e.g. Yamada [27]). 3. Linear estimates In this section we introduce several linear estimates. Thereafter we use the cut-off functions which will be used in the proofs to aligned to low-, middle- and high-frequency parts. Let χl , χm and χh ∈ C ∞ (R) be
ρ |ξ | ≤ , 1, χ (ξ ) = 2 h 0, 0, |ξ | ≥ ρ, χm (ξ ) = 1 − χl (ξ ) − χh (ξ )
χl (ξ ) =
1,
|ξ | ≥ 2R, |ξ | ≤ R,
for ρ ≪ 1 and R ≫ 1 which will be determined in each proof. We denote by δ < 1/2 a positive constant which may change from line to line. First, we show the dissipative property for K0 (t ) and K1 (t ) defined in (1.13) and (1.14). We remark that the following proposition holds in q < 2. For the solution of the damped wave equation such estimates were firstly obtained in [18]. Here, we apply the method developed by Hosono–Ogawa [9] using the Carleson–Beurling inequality. Proposition 3.1 (Lq –Lr Estimates). For any 1 ≤ r ≤ q ≤ ∞ and t ≥ 0 we have − 21
1 1 r −q
− 21
1 1 r −q
∥K0 (t )g ∥Lq (R) ≤ C (1 + t )
∥K1 (t )g ∥Lq (R) ≤ C (1 + t )
∥g ∥W 2,r (R) ,
(3.1)
∥g ∥Lr (R) .
(3.2)
Proof. We define the Fourier multiplier m1 (t , ξ ) corresponding to the evolution operator K1 (t ) by m1 ( t , ξ ) = e
− 2t
e
it
ξ 4 +αξ 2 − 14
−e
−it ξ 4 +αξ 2 − 14
2i ξ 4 + αξ 2 −
1 4
.
If we show 1
∥K1 (t )g ∥L∞ (R) ≤ C (1 + t )− 2 ∥g ∥L1 (R)
(3.3)
∥K1 (t )g ∥Lq (R) ≤ C ∥g ∥Lq (R) ,
(3.4)
and 1 ≤ q ≤ ∞,
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then we obtain (3.2) from the interpolation. We first show L∞ –L1 estimate (3.3). It follows from the Hausdorff–Young inequality that
∥K1 (t )g ∥L∞ (R) ≤ C ∥m1 (t , ·) g ∥L1 (R) ≤ C ∥m1 (t , ·)∥L1 (R) ∥ g ∥L∞ (R) ≤ C ∥m1 (t , ·)∥L1 (R) ∥g ∥L1 (R) . √ 1+α 2 2
Setting ζ (α) = − α2 +
(> 0), we see
4 ξ + αξ 2 − 1/4 ≤ 0, ξ 4 + αξ 2 − 1/4 > 0,
|ξ | ≤ ζ (α), |ξ | > ζ (α).
Then, we should regard as
1 4 − αξ 2 sinh − ξ t 4 t , e− 2 1 4 − αξ 2 − ξ 4 m1 (t , ξ ) = sin t ξ 4 + αξ 2 − 41 t −2 e , ξ 4 + αξ 2 − 1
|ξ | < ζ (α), (3.5)
|ξ | > ζ (α).
4
For a small constant ρ < ζ (α), we define the positive constant by Cρ,α =
α 1 2
+
1 4
− ρ 4 − αρ 2
.
(3.6)
We see that for |ξ | < ρ
t 14 −ξ 4 −αξ 2 −t 14 −ξ 4 −αξ 2 t sinh t 14 − ξ 4 − αξ 2 − e t e −2 e ≤ e− 2 1 4 − αξ 2 2 14 − ξ 4 − αξ 2 − ξ 4 1
1
2
4
≤ C exp t − +
− ξ 4 − αξ 2
1
≤ C exp (1 + t ) − + 2
1 4
− ξ 4 − αξ 2
≤ C exp{−Cρ,α (1 + t )ξ 2 },
(3.7)
where in the last inequality we used the fact that 1
− + 2
1 4
− ξ 4 − αξ 2 =
−(ξ 4 + αξ 2 ) ≤ + 14 − ξ 4 − αξ 2
1 2
−αξ 2 1 2
+
1 4
− ρ 4 − αρ 2
.
Therefore, by Lemma 2.3 we obtain
∥χl m1 (t , ·)∥L1 (R) ≤ C
e−Cρ,α (1+t )ξ dξ 2
|ξ |≤ρ 1
≤ C (1 + t )− 2 . Now we fix some large R > ζ (α). Then it follows that
∥χm m1 (t , ·)∥L1 (R) ≤ C
ρ/2≤|ξ |≤2R
|m1 (t , ξ )|dξ ≤ Ce−δt
(3.8)
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7
from
1 4 − αξ 2 sinh t − ξ 4 t , te− 2 1 t 4 − ξ 4 − αξ 2 |m1 (t , ξ )| ≤ t sin t ξ 4 + αξ 2 − 41 − , te 2 t ξ 4 + αξ 2 − 41
(ρ/2 ≤ |ξ | < ζ (α)),
(ζ (α) < |ξ | ≤ 2R)
t
≤ Cte− 2 ≤ Ce−δt . It also follows that
∥χh m1 (t , ·)∥L1 (R) ≤ C
t
|m1 (t , ξ )|dξ ≤ Ce− 2 ,
(3.9)
|ξ |≥R
from
sin t ξ 4 + αξ 2 − 14 1 ≤ ξ 4 + αξ 2 − 41 ξ 4 + αξ 2 −
∈ L1 (|ξ | ≥ R).
1 4
Consequently, we have
∥m1 (t , ·)∥L1 (R) ≤ ∥χl m1 (t , ·)∥L1 (R) + ∥χm m1 (t , ·)∥L1 (R) + ∥χh m1 (t , ·)∥L1 (R) 1
≤ C (1 + t )− 2 . Next, we verify Lq –Lq estimate (3.4). Recall the estimate (3.7) and Lemma 2.3, then we have
∥χl m1 (t , ·)∥L2 (R) ≤ C
e−2Cρ,α (1+t )ξ dξ 2
21
|ξ |≤ρ 1
≤ C (1 + t )− 4 . Similar computations to (3.8) and (3.9) also yield
∥χm m1 (t , ·)∥L2 (R) + ∥χh m1 (t , ·)∥L2 (R) ≤ Ce−δt .
(3.10)
Then we deduce that
∥m1 (t , ·)∥L2 (R) ≤ ∥χl m1 (t , ·)∥L2 (R) + ∥χm m1 (t , ·)∥L2 (R) + ∥χh m1 (t , ·)∥L2 (R) 1
≤ C (1 + t )− 4 .
(3.11)
We note that for |ξ | < |ζ (α)| − 2t
∂ξ {χ (ξ )m1 (t , ξ )} = −te
− 2t
+e
cosh t
1 4
sinh t
1 4
2ξ 3 + αξ
− ξ 4 − αξ 2
− ξ 4 − αξ 2
1 4
− ξ 4 − αξ 2
2ξ 3 + αξ
1 4
t
+ e− 2
sinh t
1 4
1 4
− ξ − αξ 4
2
− ξ 4 − αξ 2
χ (ξ ) 32 χ (ξ )
χ ′ (ξ ),
− ξ 4 − αξ 2
and for |ζ (α)| > |ξ |
∂ξ {χ (ξ )m1 (t , ξ )} = −te
− 2t
− 2t
+e
cos t
ξ + αξ − 4
2
sin t
ξ + αξ − 4
2
1
4 1 4
2ξ 3 + αξ
ξ 4 + αξ 2 −
1 4
2ξ 3 + αξ
ξ 4 + αξ 2 −
1 4
χ (ξ ) 32 χ (ξ )
8
H. Takeda, S. Yoshikawa / J. Math. Anal. Appl. (
sin t
t
+ e− 2
ξ 4 + αξ 2 −
ξ + αξ − 4
2
)
–
1 4
χ ′ (ξ ).
1 4
We shall use the above expression of ∂ξ {χ (ξ )m1 (t , ξ )} by replacing χ to χl , χm and χh . Recalling ρ < ζ (α), from (3.7) we have for |ξ | ≤ ρ
|∂ξ {χl (ξ )m1 (t , ξ )}| ≤ Ct |ξ |e−Cρ,α (1+t )ξ χl (ξ ) + C |ξ |e−Cρ,α (1+t )ξ χl (ξ ) + Ce−Cρ,α (1+t )ξ |χl′ (ξ )|, 2
2
2
(3.12)
where Cρ,α was defined in (3.6). Then we have
2 ∥∂ξ {χl m1 (t , ·)}∥L2 (R) ≤ Ct ξ e−Cρ,α (1+t )ξ χl 2
2 + C ξ e−Cρ,α (1+t )ξ χl 2
L (R)
−3/4
≤ Ct (1 + t )
1/4
≤ C (1 + t )
L (R)
−3/4
+ C (1 + t )
2 + C ξ e−Cρ,α (1+t )ξ χl′ 2
L (R)
−δ(1+t )
+ Ce
,
(3.13)
where we used Lemma 2.3 and the fact that
∥e−Cρ,α (1+t )ξ χl′ ∥L2 (R) ≤ ∥e−Cρ,α (1+t )ξ ∥L2 (ρ/2≤|ξ |≤ρ) 2
2
≤ e−Cρ,α (1+t ) −δ(1+t )
≤ Ce
ρ2 4
∥1∥L2 (ρ/2≤|ξ |≤ρ)
.
Since for ν(ξ ) = ξ + αξ − 1/4 4
2
√ √ √ (t −ν)cosh(t −ν) − sinh(t −ν) t , √ e− 2 (2ξ 3 + αξ )t 3 3 √(t −ν) √ √ ∂ξ m1 (t , ξ ) = (t ν) cos(t ν) − sin(t ν) t e− 2 (2ξ 3 + αξ )t 3 , √ (t ν)3
|ξ | < ζ (α), |ξ | > ζ (α),
and lim
y coshy − sinhy
y→0
y3
= lim
y cos y − sin y
y→0
y3
1
=− , 3
we have
∥∂ξ m1 (t , ·)∥L∞ (ρ/2≤|ξ |≤2R) ≤ Ce−δt . Then we obtain
∥∂ξ {χm m1 (t , ·)}∥L2 (R) ≤ Ce−δt .
(3.14)
For the high frequency part, it follows that
2ξ 3 + αξ |∂ξ {χh m1 (t , ξ )}| ≤ te 4 χ (ξ ) h ξ + αξ 2 − 41 3 2 ξ + αξ 1 t t −2 ′ −2 + e χh (ξ ) + e χh (ξ ) 3 1 ξ 4 + αξ 2 − 1 2 ξ 4 + αξ 2 − 4 4 ξ3 ξ3 t − 2t −δ t ≤ Ce 4 χ (ξ ) + Ce χh (ξ ) + Ce− 2 |χh′ (ξ )|. 3 ξ + αξ 2 − 14 h ξ 4 + αξ 2 − 1 2 4 − 2t
Note that
ξ3 ξ 4 + αξ 2 −
1 4
∈ L2 (R, ∞),
ξ3
2 3 ∈ L (R, ∞) ξ 4 + αξ 2 − 41 2
and suppχh′ (ξ ) ⊂ [R, 2R]. Therefore, we deduce that
∥∂ξ {χh m1 (t )}∥L2 (R) ≤ Ce−δt .
(3.15)
H. Takeda, S. Yoshikawa / J. Math. Anal. Appl. (
)
–
9
Combining the estimates (3.13)–(3.15) yields
∥∂ξ m1 (t , ·)∥L2 (R) ≤ ∥∂ξ {χl m1 (t , ·)}∥L2 (R) + ∥∂ξ {χm m1 (t , ·)}∥L2 (R) + ∥∂ξ {χh m1 (t , ·)}∥L2 (R) 1
≤ C (1 + t ) 4 .
(3.16)
It follows from Lemma 2.1 with (3.11) and (3.16) that for q ∈ [1, ∞] 1
1
Mq (m1 (t , ·)) ≤ C ∥m1 (t , ·)∥L22 (R) ∥∂ξ m1 (t , ·)∥L22 (R) 1
1
≤ C (1 + t ) 8 (1 + t )− 8 = C , where C is independent of t. Therefore, we are led to (3.4):
∥K1 (t )g ∥Lq (R) = ∥F −1 [m1 g ]∥Lq (R) ≤ C ∥g ∥Lq (R) for any q ∈ [1, ∞]. This completes the proof of (3.2). For the proof of (3.1), recalling that
K0 (t )g = F −1 e
− 2t
e
it
−it ξ 4 +αξ 2 − 41
ξ 4 +αξ 2 − 14
+e 2(1 + ξ 2 )
(1 + ξ 2 ) g
and employing the Fourier multiplier m0 ( t , ξ ) = e
− 2t
e
it
−it ξ 4 +αξ 2 − 14
ξ 4 +αξ 2 − 14
+e 2(1 + ξ 2 )
,
we can prove easily from a modification similar to the above for m0 instead of m1 , then we omit the proof.
To deal with the nonlinear term of (1.1) including derivative, we use the following estimates with the second order derivative. We use the stationary phase method by Van der Corput to estimate the high frequency part. Proposition 3.2 (Lq –Lr Estimates With Second Order Derivative). For 2 ≤ q ≤ ∞, 1 ≤ r ≤ q ≤ ∞ such that t > 0, it follows that − 21
− 12
∥∂x2 K0 (t )g ∥Lq (R) ≤ C (1 + t ) ∥∂x2 K1 (t )g ∥Lq (R) ≤ C (1 + t )
1 1 r −q
−1
1 1 r − q −1
t
∥g ∥W 2,r (R) + Ce− 2 t t
∥g ∥Lr (R) + Ce− 2 t
− 12 1− 2q
− 12 1− 2q
∥g ∥W 2,q′ (R) ,
∥g ∥Lq′ (R) .
1 q
+
1 q′
= 1 and
(3.17) (3.18)
Proof. We decompose the linearized solution to three parts, K1 (t )g = F −1 [(χl + χm + χh )m1 (t , ·) g]
=: K1l (t )g + K1m (t )g + K1h (t )g , where K1j (t )g = F −1 [χj m1 (t , ·) g ] for j = l, m, h. We will show
∥∂
2 x K1l
(t )g ∥Lq (R) ≤ C (1 + t )
2 x K1m
∥∂
− 21
(t )g ∥Lq (R) ≤ Ce
−δ t t
1 1 r −q
−1
∥g ∥Lr (R) ,
(3.19)
∥g ∥Lr (R) ,
∥∂x2 K1h (t )g ∥Lq (R) ≤ Ce− 2 t
− 21 1− 2q
(3.20)
∥g ∥Lq′ (R) .
(3.21)
The estimate (3.19) is derived by the same way as the proof of Proposition 3.1. From (3.7) and Lemma 2.3
2 ∥∂x2 K1l (t )g ∥L∞ (R) ≤ C ξ 2 e−Cρ,α (1+t )ξ 1
L (|ξ |≤ρ)
∥g ∥L1 (R)
3
≤ C (1 + t )− 2 ∥g ∥L1 (R) . q
(3.22)
q
Next, we prove the L –L estimate of (3.19). It follows from the same computation as the above that
∥ξ 2 m1 (t , ·)χl ∥L2 (R) ≤ C
∞
e−2Cρ,α (1+t )ξ ξ 4 dξ 2
−∞ 5
≤ C (1 + t )− 4 ,
21
10
H. Takeda, S. Yoshikawa / J. Math. Anal. Appl. (
)
–
and from (3.7) and (3.12) that
∥∂ξ {ξ 2 m1 (t , ·)χl }∥L2 (R) ≤ 2∥ξ m1 (t , ·)χl ∥L2 (R) + ∥ξ 2 ∂ξ {m1 (t , ·)χl }∥L2 (R) ≤ C ∥ξ e−Cρ,α (1+t )ξ ∥L2 (R) 2 2 2 + C t ∥ξ 3 e−Cρ,α (1+t )ξ ∥L2 (R) + ∥ξ 3 e−Cρ,α (1+t )ξ ∥L2 (R) + ∥ξ 2 e−Cρ,α (1+t )ξ χl′ ∥L2 (R) 2
3
7
7
≤ C (1 + t )− 4 + C {t (1 + t )− 4 + (1 + t )− 4 + e−δt } 3
≤ C (1 + t )− 4 . Then by Lemma 2.1 we deduce that 1/2
1/2
Mq (ξ 2 m1 (t , ·)χl ) ≤ C ∥ξ 2 m1 (t , ·)χl ∥L2 (R) ∥∂ξ {ξ 2 m1 (t , ·)χl }∥L2 (R) 5
3
≤ C (1 + t )− 8 − 8 = C (1 + t )−1 . Therefore, we arrive at
∥∂x2 K1l (t )g ∥Lq (R) ≤ C (1 + t )−1 ∥g ∥Lq (R)
(3.23)
for any q ∈ [1, ∞]. The interpolation between (3.22) and (3.23) yields (3.19). Next, we prove (3.20). Since (ρ/2) ≤ |ξ |2 ≤ (2R)2 , we have 2
∥ξ 2 m1 (t , ·)χm ∥L1 (R) ≤ C ∥m1 (t , ·)χm ∥L1 (R) ≤ Ce−δt , ∥ξ 2 m1 (t , ·)χm ∥L2 (R) ≤ C ∥m1 (t , ·)χm ∥L2 (R) ≤ Ce−δt , ∥∂ξ {ξ 2 m1 (t , ·)χm }∥L2 (R) ≤ C (∥m1 (t , ·)χm ∥L2 (R) + ∥∂ξ {m1 (t , ·)χm }∥L2 (R) ) ≤ Ce−δt , by virtue of (3.8), (3.10) and (3.14), and then
∥∂x2 K1m (t )g ∥L∞ (R) ≤ Ce−δt ∥g ∥L1 (R) ,
∥∂x2 K1m (t )g ∥Lq (R) ≤ Ce−δt ∥g ∥Lq (R) ,
which give the inequality (3.20). Finally we prove the estimate (3.21). By the Plancherel theorem, we see
2 −t ξ χ h 2 g e =C 1 4 2 ξ + αξ − 4 2
∥∂x2 K1h (t )g ∥L2 (R)
t
≤ Ce− 2 ∥g ∥L2 (R) .
L (R)
Thus the proof of (3.21) is reduced to verifying the inequality 1
t
∥∂x2 K1h (t )g ∥L∞ (R) ≤ Ce− 2 t − 2 ∥g ∥L1 (R) . To show (3.24), we first decompose ∂
2 x K1h
e
it
(t )g into two parts,
ξ 4 +αξ 2 − 14
t ∂x2 K1h (t )g = −e− 2 F −1
= −e
−it ξ 4 +αξ 2 − 14
−e
2i ξ 4 + αξ 2 − it
χh ξ 2
1 4
e
ξ 4 +αξ 2 − 41
t
t
= −e− 2 J+ (t , ·) ∗ g + e− 2 J− (t , ·) ∗ g , where we set ±it ξ 4 +αξ 2 − 41
J± (t , x) = F −1
e
2i ξ 4 + αξ 2 −
1
= (2π )− 2
g
−it ξ 4 +αξ 2 − 14 2 2 χ ξ e χ ξ t h h F −1 g + e − 2 F −1 g 2i ξ 4 + αξ 2 − 41 2i ξ 4 + αξ 2 − 41
− 2t
(3.24)
e R
ixξ ±it
1 4
ξ 2 χh
ξ 4 +αξ 2 − 14
ξ 2 χh (ξ ) dξ . 2i ξ 4 + αξ 2 − 41
H. Takeda, S. Yoshikawa / J. Math. Anal. Appl. (
)
–
11
To apply Lemma 2.2 to J± as
φ(ξ ) =
xξ t
1
ξ 4 + αξ 2 − ,
±
ψ(ξ ) =
4
ξ 2 χh (ξ )
2i ξ 4 + αξ 2 −
1 4
,
we check that φ and ψ satisfy the assumptions of Lemma 2.2. We immediately have ψ ∈ L∞ (R) and
∂ξ ψ(ξ ) =
(2αξ 3 − ξ )χh (ξ ) ξ 2 χh′ (ξ ) + 3 / 2 4i ξ 4 + αξ 2 − 14 2i ξ 4 + αξ 2 −
∈ L1 (R, ∞).
1 4
Further, the direct computation yields
′′ |φ (ξ )| = ∂ξ2 ξ 4 + αξ 2 −
2ξ 6 + 3αξ 4 − 3ξ 2 /2 − α/4 . = 3 4 ξ 4 + αξ 2 − 1 2 1
4
Choosing sufficiently large R ≫ 1 such that 3αξ − 3ξ /2 − α/4 > 0 and ξ 4 + αξ 2 − 1/4 < 3ξ 4 /2 for |ξ | ≥ R, we have 4
2ξ 6
|φ ′′ (ξ )| > 3
2
ξ4
2
2
23 = 3 32 > 1. 2
Then, applying Lemma 2.2 for arbitrary M ≫ R, we obtain
M
e (
it xt ξ ±φ(ξ )
R
M ′ ) ψ(ξ )dξ ≤ 10|t |−1/2 |ψ(M )| + |ψ (ξ )| d ξ R ∞ −1/2 ′ ≤ C |t | ∥ψ∥L∞ (R) + |ψ (ξ )|dξ R
≤ C |t |−1/2 . Since the above C is independent of M, tending M → ∞ yields
∞
R
eit ( t ξ ±φ(ξ )) ψ(ξ )dξ ≤ C |t |− 2 . x
1
In the same fashion, we also have
−R
e (
it xt ξ ±φ(ξ )
−∞
) ψ(ξ )dξ ≤ C |t |− 21 .
Combining these, we obtain 1
|J± (t , x)| ≤ C |t |− 2 . Therefore, it follows from the Hausdorff–Young inequality that 1
∥J± (t ) ∗ g ∥L∞ (R) ≤ C ∥J± (t )∥L∞ (R) ∥g ∥L1 (R) ≤ C |t |− 2 ∥g ∥L1 (R) , which completes the proof of (3.24). Consequently, the desired estimate (3.18) is obtained. We omit the proof of (3.17), since the proof is similar to the above. To deal with the nonlinear part without regularity loss we shall need the following estimate. The proof is essentially the same as that of Proposition 3.1. Then we omit it. Corollary 3.3. For 1 ≤ r ≤ q ≤ ∞, 2 ≤ q ≤ ∞ and t > 0, it follows that − 12
∥∂x3 K1 (t )g ∥Lq (R) ≤ C (1 + t )
1 1 r −q
− 23
t
∥g ∥Lr (R) + Ce− 2 t
− 12 1− 1q
∥∂x g ∥Lq′ (R) .
4. Global existence In this section, we show the global existence of solution for (1.12). Our proof is based on the contraction mapping principle with the help of the linear estimates introduced in Section 3.
12
H. Takeda, S. Yoshikawa / J. Math. Anal. Appl. (
)
–
Proof of Theorem 1.1. Let us introduce the mapping
Ψ [u](t ) = K (g0 , g1 ) +
t
K1 (t − s)∂x f (∂x u(s))ds, 0
for K (g0 , g1 ) = K0 (t )g0 + K1 (t )
1 2
g0 + g1
,
and the complete metric space
1 2 X = u(t ) ∈ C [0, ∞); L (R) ∩ H (R) ; ∥u∥X ≤ N , with
∥u∥X = sup
t ∈[0,∞)
5 ∥u(t )∥L1 (R) + (1 + t ) 4 ∥∂x2 u(t )∥L2 (R) ,
where N is determined later. We shall prove the mapping Ψ is contraction in X . Thus we concentrate on showing the following claims: (I) ∥Ψ [u]∥X ≤ N for u ∈ X . (II) ∥Ψ [u] − Ψ [v]∥X ≤ 12 ∥u − v∥X for u, v ∈ X . Setting
δ0 = ∥g0 ∥W 2,1 (R)∩H 2 (R) + ∥g1 ∥L1 (R)∩L2 (R) , by (3.1) and (3.2), we have
∥K (g0 , g1 )∥L1 (R) ≤ C ∥g0 ∥W 2,1 (R) + ∥g1 ∥L1 (R) ≤ C δ0 .
(4.1)
It follows from the Gagliardo–Nirenberg inequality (see e.g. [3, Theorem 1.3.7]) that
∥∂x u∥L2(p−1) (R) ≤ C ∥u∥ 1
1 1 1+ p− 5 1 L1 (R)
1 4 4− p− 5 1 2 x u L2 (R)
,
∥∂ ∥
(4.2)
4
∥∂x u∥L∞ (R) ≤ C ∥u∥L51 (R) ∥∂x2 u∥L52 (R) .
(4.3)
Applying (4.2), we obtain
∥∂x f (∂x u(s))∥L1 (R) ≤ C f ′ (∂x u(s))∂x2 u(s)L1 (R) ≤ C ∥∂x u(s)∥pL2−(p1−1) (R) ∂x2 u(s)L2 (R) p 4p ≤ C ∥u(s)∥ 51 ∂ 2 u(s) 52 L (R)
L (R)
x
p 5 L1 (R)
≤ C (1 + s)−p ∥u(s)∥
4p5 5 (1 + s) 4 ∂x2 u(s)L2 (R)
≤ C (1 + s)−p ∥u∥pX , which gives t
0
∥K1 (t − s)∂x f (∂x u(s))∥L1 (R) ds ≤ C
t
∥∂x f (∂x u(s))∥L1 (R) ds t ≤ C ∥u∥pX (1 + s)−p ds 0
0 p X
≤ C ∥ u∥ .
(4.4)
Combining the estimates (4.1) and (4.4), we have
∥Ψ [u](t )∥L1 (R) ≤ ∥K (g0 , g1 )∥L1 (R) + ≤ C δ0 + C ∥u∥pX .
t
0
∥K1 (t − s)∂x f (∂x u(s))∥L1 (R) ds (4.5)
H. Takeda, S. Yoshikawa / J. Math. Anal. Appl. (
)
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13
From (4.3) we obtain
−1 2 ∥∂x f (∂x u(s))∥L2 (R) ≤ C ∥∂x u(s)∥pL∞ (R) ∂x u(s) L2 (R) p−1 4(p−1) +1 ≤ C ∥u(s)∥ 15 ∂ 2 u(s) 2 5 L (R)
L (R)
x
1
≤ C (1 + s)−p− 4 ∥u(s)∥
p 5 L1 (R)
4p5 + 15 5 (1 + s) 4 ∂x2 u(s)L2 (R)
1
≤ C (1 + s)−p− 4 ∥u∥pX .
(4.6)
It then follows from (3.1) and (3.2) that
2 ∂ K (g0 , g1 ) 2
L (R)
x
t 5 ≤ C (1 + t )− 4 ∥g0 ∥W 2,1 (R) + ∥g1 ∥L1 (R) + Ce− 2 ∥g0 ∥H 2 (R) + ∥g1 ∥L2 (R) 5
≤ C (1 + t )− 4 δ0 and that t
0
t
t t −s 5 (1 + t − s)− 4 ∥∂x f (∂x u(s))∥L1 (R) ds + C e− 2 ∥∂x f (∂x u(s))∥L2 (R) ds 0 0 t t t −s 1 p p − 54 −p ≤ C ∥ u∥ X (1 + t − s) (1 + s) ds + C ∥u∥X e− 2 (1 + s)−p− 4 ds 0 0 5 1 ≤ C (1 + t )− 4 + (1 + t )−p− 4 ∥u∥pX
2 ∂ K1 (t − s)∂x f (∂x u(s)) 2 ds ≤ C x L (R)
5
≤ C (1 + t )− 4 ∥u∥pX , by virtue of Lemma 2.4. Therefore, we have
∥∂x2 Ψ [u](t )∥L2 (R) ≤ ∂x2 K (g0 , g1 )L2 (R) + 5
t
2 ∂ K1 (t − s)∂x f (∂x u(s)) 2
L (R)
x
0 5
≤ C (1 + t )− 4 δ0 + C (1 + t )− 4 ∥u∥pX .
ds (4.7)
Consequently, the estimates (4.5) and (4.7) show for constants C0 and C
∥Ψ [u]∥X ≤ C0 δ0 + C ∥u∥pX ≤ C0 δ0 + CN p . In a similar fashion, it can be shown that
∥Ψ [u] − Ψ [v]∥X ≤ C (∥u∥pX−1 + ∥v∥pX−1 )∥u − v∥X ≤ 2CN p−1 ∥u − v∥X . Let us define N by N = 2C0 δ0 . Choosing δ0 sufficiently small satisfying 4CN p−1 < 1, we obtain the claim (I) and (II). Thereby, from the Banach fixed point theorem, there exists a unique fixed point u ∈ X such that u(t ) = Ψ [u](t ), which completes the proof.
In Theorem 1.1 the persistence property of solution does not hold because u ̸∈ C ([0, ∞); W 2,1 (R) ∩ H 2 (R)). However, in an argument similar to the proof of Theorem 1.1 we can also obtain the global well-posedness result for H 2 -solution. Corollary 4.1 (Global Well-Posedness). Assume that the nonlinear term f satisfies the assumptions (1.3). If the initial data (g0 , g1 ) ∈ H 2 (R) × L2 (R) and
∥g0 ∥H 2 (R) + ∥g1 ∥L2 (R) is sufficiently small, then there exists a unique mild solution u(t ) for the initial value problem (1.1)–(1.2) in the class C ([0, ∞); H 2 (R)) satisfying
∥u(t )∥L2 (R) ≤ C ,
∥∂x2 u(t )∥L2 (R) ≤ C (1 + t )−1 ,
t ≥ 0.
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)
–
5. Smoothing of the solution In the previous section, we construct the time-global solution for the integral equation (1.12). In this section, we investigate the smoothing effect of the solution. Proof of Theorem 1.3. For 2 ≤ q ≤ ∞, we have
∥∂x2 u(t )∥Lq (R) ≤ ∥∂x2 K (g0 , g1 )∥Lq (R) +
t
∥∂x2 K1 (t − s)∂x f (∂x u(s))∥Lq (R) ds. 0
From Proposition 3.2 we have − 21 1− 1q −1
∥∂x2 K (g0 , g1 )∥Lq (R) ≤ C (1 + t )
≤ Ct
− 12 1− 1q
−1
t
∥(g0 , g1 )∥W 2,1 ×L1 (R) + e− 2 t
− 12 1− 2q
∥(g0 , g1 )∥W 2,1 (R)∩H 2 (R)×L1 (R)∩L2 (R)
∥(g0 , g1 )∥W 2,1 (R)∩H 2 (R)×L1 (R)∩L2 (R) .
We shall show that the nonlinear part decays more rapid than the linear part. From Proposition 3.2 and Corollary 3.3 we have t
∥∂
2 x K1
(t − s)∂x f (∂x u(s))∥
t 2
Lq (R)
∥∂x3 K1 (t − s)f (∂x u(s))∥Lq (R) ds
ds ≤ 0
0
t
∥∂x2 K1 (t − s)∂x f (∂x u(s))∥Lq (R) ds
+ t 2 t 2
− 21 1− 1q − 32
(1 + t − s)
≤C 0
t
e−
+C
t −s 2
∥f (∂x u(s))∥L1 (R) ds
− 12 1− 2q
( t − s)
0 t
+C t 2
(1 + t − s)
− 12
1 − 1q 2
∥∂x f (∂x u(s))∥Lq′ (R) ds
−1
∥∂x f (∂x u(s))∥L2 (R) ds.
(5.1)
It follows from the Gagliardo–Nirenberg inequality (see e.g. [3, Theorem 1.3.7]) that for any r ∈ [4/3, ∞]
1− 25 2− 1r
∥∂x u(s)∥Lr (R) ≤ C ∥u(s)∥L1 (R) ≤ C (1 + s)
2 2− 1r 5
∥∂x2 u(s)∥L2 (R)
− 21 1− 1r − 12
,
(5.2)
with the help of (1.15). Then, we obtain t 2
− 21 1− 1q − 32
(1 + t − s)
0
∥f (∂x u(s))∥L1 (R) ds ≤ C 1 +
t
− 12
1− 1q − 32
t 2
2
0
≤ C (1 + t )
− 12 1− 1q
− 32
≤ C (1 + t )
− 12 1− 1q
− 32
t 2
∥∂x u(s)∥pLp (R) ds 1
(1 + s)−p+ 2 ds
0 t 2
3
(1 + s)− 2 ds
0
≤ C (1 + t )
− 12 1− 1q − 32
,
where C is independent of t. Next, we calculate the second term in the right hand side of (5.1). Since 1 ≤ q′ ≤ 2, we have t
e
s − t− 2
− 21 1− 2q
(t − s)
0 t
e−
≤C
t −s 2
∥∂x f (∂x u(s))∥Lq′ (R) ds ≤ C
− 21 1− 2q
(t − s)
1
(1 + s)−p− 2q ds
0 1
≤ C (1 + t )−p− 2q ≤ C (1 + t )
− 12 1− 1q − 32
,
t
s − t− 2
e 0
( t − s)
− 12 1− 2q
∥∂x u(s)∥p−2q1 L q−2
(p−1)
(R)
∥∂x2 u(s)∥L2 (R) ds
H. Takeda, S. Yoshikawa / J. Math. Anal. Appl. (
)
–
15
by virtue of the Holder inequality, the estimate (5.2) and Lemma 2.4(ii). Lastly, the third term of the right hand side of (5.1) can be estimated by t
t 2
(1 + t − s)
− 21
1 − 1q 2
−1
∥∂x f (∂x u(s))∥L2 (R) ds ≤ C
t
t 2
(1 + t − s)
− 12
1 − 1q 2
−1
1
(1 + s)−p− 4 ds
1
≤ C (1 + t )−p− 4 log(2 + t ) − 12 1− 1q − 32
≤ C (1 + t )
− 12 1− 1q − 32
≤ C (1 + t ) due to (4.6). This completes the proof of Theorem 1.3.
1
1
{(1 + t )− 2q − 4 log(2 + t )} ,
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