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The orbital stability of the solitary wave solutions of the generalized Camassa–Holm equation
J. Math. Anal. Appl. 398 (2013) 776–784
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The orbital stability of the solitary wave solutions of the generalized Camassa–Holm equation✩ Xiaohua Liu a , Weiguo Zhang b,∗ , Zhengming Li b a
Guizhou Minzu University, GuiYang 550025, PR China
b
University of Shanghai for Science and Technology, ShangHai 200093, PR China
article
info
Article history: Received 27 April 2010 Available online 23 September 2012 Submitted by Thomas P. Witelski Keywords: Generalized Camassa–Holm equation Stability Solitary wave solution
abstract In this paper, we consider the orbital stability of smooth solitary wave solutions of the generalized Camassa–Holm equation. By constructing the functional extremum problem and using the orbital stability theory presented by Grillakis, Shatah, Strauss and Bona, and Souganidis, we show that the solitary wave solutions of the generalized Camassa–Holm equation are orbitally stable or unstable as determined by the sign of a discriminant. The conclusions presented by the previous authors, such as Hakkaev and Kirchev, Constantin and Strauss, can be considered as a special case of our results. © 2012 Elsevier Inc. All rights reserved.
1. Introduction In 1993, Camassa and Holm [1] derived the shallow water wave equation ut + 2kux − uxxt + 3uux = 2ux uxx + uuxxx .
(1)
Eq. (1) is called the Camassa–Holm (CH) equation, which models the unidirectional propagation of water waves in the shallow water regime, when the wavelength is considerably larger than the average water depth. Where u denotes the fluid velocity, or can also be interpreted as the height of the water’s free surface above a flat bottom, and the constant k is related to the critical shallow water wave speed. For the special case k = 0, they [1] showed that (1) has the peaked solitary waves u(x, t ) = c exp(−|x − ct |), which were called peakons due to the discontinuity of the first derivation at the wave vertex; By applying the decomposition method to Eq. (1), three approximate solutions were obtained [2], which are similar to the peaked solitary waves. The CH equation has been investigated [3–10] extensively. Boyd [7] investigated the peaked periodic waves that have a discontinuous first derivative at each peak called coshoidal waves or periodic cusp waves. The analytic expressions of peaked solitary waves and peaked periodic waves of CH Eq. (1) were obtained in [8] by using the bifurcation method of planar dynamical systems. The orbital stability of smooth solitary wave solutions of Eq. (1) is considered by Constantin and Strauss [9], and the stability of the non-smooth peakons (k = 0) by Constantin [4] and Constantin [10] by means of variational methods.
✩ This project is supported by National Natural Science Foundation of China (No. 11071164), Scientific Research Innovation Project of Shanghai Municipal Education Commission (No. 13ZZ118), Shanghai Natural Science Foundation (No. 10ZR1420800) and Leading Academic Discipline Project of Shanghai Municipal Government. ∗ Corresponding author. E-mail address: [email protected] (X. Liu).
0022-247X/$ – see front matter © 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2012.09.014
X. Liu et al. / J. Math. Anal. Appl. 398 (2013) 776–784
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If the nonlinear convection term uux of Eq. (1) is changed into the new nonlinear convection term um ux , we have the following modified Camassa–Holm(MCH) equation ut + 2kux − uxxt + aum ux = 2ux uxx + uuxxx .
(2)
Because of the new nonlinear convection term, the Eq. (2) creates many new nonlinear phenomena, such as compacton solitons with compact support, solitons with cusps, or peakons [11–18]. Tian and Yin [11] proposed four simple ansätzs to obtain abundant solutions: compactons, solitary patterns solutions having infinite slopes or cusps, and solitary waves. Qian and Tang [16] studied the peakons and the periodic cusp wave solution of the MCH Eq. (2), they constructed some smooth periodic wave solutions, periodic cusp wave solutions, and oscillatory solitary wave solutions. Liu and Qian [17] investigated the peakons and their bifurcation of the MCH Eq. (2), and for special cases they gave some explicit expressions for the peakons. Tian and Song [18] derived some new exact peaked solitary waves. Khuri [19] studied the periodic wave and peaked solitary waves of Eq. (2) by employing polynomial ansätz. Considered herein is the stability of the solitary wave solutions to the generalized Camassa–Holm (GCH) equation
ut − uxxt +
2ku +
p+2 2
u
p+1
= pu
2 p−1 ux
2
x
p
,
+ u uxx
(3)
x
where k ∈ R, p > 0. The GCH Eq. (3) reduces to the CH Eq. (1) for p = 1. We know that the Eq. (3) has more new nonlinear phenomena and structures than the Eqs. (1) and (2). For any p, the explicit or implicit expression of traveling wave solutions to Eq. (3) will have become more and more complex, but so far have not been presented in papers. Solitary waves are valuable in theory and practice, in applications when they are stable in some sense. It is significant to study the orbital stability of solitary waves. There exist a few papers on the stability of solitary waves of Eq. (3). Hakkev [20] establish the orbital stability of smooth solitary wave solutions of Eq. (3) by virtue of the orbital stability theory presented by Grillakis–Shatah–Strauss [21], and the local well posedness of the Cauchy problem of Eq. (3) in the Sobolev space H s (s > 23 ) by applying the method of the pseudoparabolic regularization. Hakkev [22] discussed the stability of non-smooth peakons of Eq. (3) by the method employed by Constantin [4], and the well posedness by using Kato’s semigroup approach. In this paper, the main results are represented in Section 2. In Section 3, the stability of solitary wave solutions of Eq. (3) is studied with the help of the orbital stability theory [21]. In Section 4, the instability of solitary waves of Eq. (3) is considered by using the theory reported by [23]. Then we can deduce the results of [20] using the analysis done here. Specially, by using the method of the third and fourth section, we can establish the proof of Theorem 4.4 in [20]. 2. Main results and notation Notation: Let Ls = Ls (R) be the Lebesgue measurable space with
|f (x)| dx s
∥f ∥Ls =
1s
,
R
and H s = H s (R) be the Sobolev space with
2 (1 + |ξ |2 )s gˆ (ξ ) dξ
∥g ∥H s =
21
,
R
where gˆ (ξ ) = R e−iξ x g (x)dx. For any Banach space X and C ([0, T ], X ) is the set of all continuous functions from [0, T ] to X . Eq. (3) can be rewritten in a Hamiltonian form
du dt
= JE ′ (u),
u ∈ X,
(4)
where X = H 2 , whose dual space is denoted by X ∗ = H −2 , τ = 1 − ∂x2 . J = τ −1 ∂x is a skew-symmetric linear operator. The Hamiltonian is E (u) = −
2
ku +
up+2
R
2
+
up 2
u2x
dx.
(5)
It follows (5) that E ′ (u) = −2ku − E ′′ (u) = −2k −
p+2 2
up+1 +
p 2
(p + 2)(p + 1) 2
up−1 u2x + up uxx ,
up +
Let the inner product of X be (u, v) =
p(p − 1) 2
up−2 u2x + pup−1 uxx + pup−1 ux ∂x + up ∂x2 .
(6) (7)
(uv + ux vx + uxx vxx )dx. For any u, v ∈ X , there exists a natural isomorphism 4 2 I : X → X defined by⟨Iu, v⟩ = (u, v), where ⟨u, v⟩ = R uv dx, I = ∂∂x4 − ∂∂x2 + 1. Let T be a one parameter group of unitary ∗
R
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X. Liu et al. / J. Math. Anal. Appl. 398 (2013) 776–784
operator on X defined by T (s)u(·) = u(· − s), s ∈ R, u ∈ X . So T ′ (0) = − ∂∂ and JB = T ′ (0), then B = −(1 − ∂x2 ). Defining x the functional 1 1 Q (u) = − ⟨Bu, u⟩ = − 2 2
(u2 + u2x )dx.
(8)
Q ′′ (u) = 1 − ∂x2 .
(9)
R
It follows (8) that Q ′ (u) = −Bu = u − uxx ,
Defining orbital φ(· − y) = {T (y)φ(·)|y ∈ R}. They may be interpreted physically as ‘‘solitary waves’’ or ‘‘bound states’’. Definition 2.1. The orbital φ(· − y) is stable if for any ε > 0, there exists δ > 0 with the following property. If ∥u(·, 0) − φ(·)∥H 2 < δ , for some T > 0, u ∈ C ([0, T ], H 2 ), and u is a solution of Eq. (3) with u(·, 0) = u0 , then u can be continued to a solution in 0 ≤ T < ∞ and sup inf ∥u (·, t ) − φ (· − y)∥H 2 < ε. t
y
Otherwise, the orbital φ(· − y) is called unstable. Definition 2.1 can be seen in Refs. [20,21,23]. Substituting u(x, t ) = φ(x − ct ) = φ(ξ ) into Eq. (3), integrating once with respect to ξ , taking integral constant zero, Eq. (3) becomes c φ − c φ ′′ − 2kφ −
p+2 2
φ p+1 + pφ p−1
φ′2 2
+ φ p φ ′′ = 0.
(10)
By (6) and (9), Eq. (10) can be rewritten as E ′ (φ) + cQ ′ (φ) = 0, φ is called a bound-state solution. To study the orbital stability of the solitary wave solutions of Eq. (3), we need operator Hc = E ′′ (φ) + cQ ′′ (φ) and function d(c ) = E (φ) + cQ (φ). From (7) and (9), we have Hc = ∂x (φ p − c )∂x + c − 2k −
(p + 2)(p + 1) 2
φp +
p(p − 1) 2
2
φ p−2 φ ′ + pφ p−1 φ ′′ .
(11)
First, we give the main results: Theorem 2.2. Suppose that φ(· − y) is a solitary wave solution, for c > 2k, if d′′ (c ) > 0, then φ(· − y) is stable. Theorem 2.3. Suppose that φ(· − y) is a solitary wave solution, for c > 2k, if d′′ (c ) < 0, then φ(· − y) is unstable. Remark 2.4. By applying Theorems 2.2 and 2.3 to the smooth solitary wave solutions to Eq. (3), we can yield Theorem 4.4 and Theorem 4.5 in [20] and Theorem in [9](p = 1). 3. Proof of Theorem 2.2 To give the proof of Theorem 2.2, we need the following lemma: Lemma 3.1. Suppose that u0 ∈ H s , s > 23 , Then there is a real number T > 0 and a unique solution u to the initial-value problem of Eq. (3) such that u ∈ C ([0, T ), H s ) satisfying u(x, 0) = u0 . Proof. Proof see Theorem 3.4 in [20].
Lemma 3.2. Let u be a solution of Eq. (3) satisfying u(x, 0) = u0 , then E (u) = E (u0 ), Q (u) = Q (u0 ), and I (u) = I (u0 ) =
u0 (x)dx
R
convergence. Proof. Integrating Eq. (4) over x ∈ [a, b], t ∈ [0, t ], we have
t
b
t
b
ut dxdt = 0
a
0
p + 2 p+1 p τ −1 ∂x −2ku − u + up−1 u2x + up uxx dxdt .
a
2
2
Letting a → +∞, b → −∞, it follows that u and its derivation decay to zero at infinity such that I (u) = I (u0 ) =
u0 (x)dx
R
converge. For proofs of the first and second equalities, we refer the readers to Ref. [20].
X. Liu et al. / J. Math. Anal. Appl. 398 (2013) 776–784
Lemma 3.3. Let u0 ∈ H s (s >
sup −∞
z
−∞
779
), and u(x, t ) is a solution of Eq. (3) satisfying u(x, 0) = u0 . Then 3 u(x, t )dx ≤ r0 (1 + t ) 4 , 3 2
(12)
where r0 depends on u0 . Proof. See Theorem 4.1 of [20] or Theorem 2.2 of [23]. Lemma 3.4. For each c ∈ (2k, ∞), Hc has a unique simple negative eigenvalue, zero is a simple eigenvalue, and the rest of its spectrum is bounded away from zero. Proof. For any u, v ∈ X , we have
φ uxx v dx = p
R
pφ
p−1
R
uxx v dx =
φx vx udx +
R
R
uvxx dx and
φ vxx udx − p
R
pφ p−1 φx ux v dx. R
Therefore,
p ∂x φ − c ∂x u, v = ∂x φ p − c ∂x v, u . By (11), we know that Hc is a self-conjugate operator and Hc φ ′ = (φ p − c )φ ′′′ + 2pφ p−1 φ ′ φ ′′ + (c − 2k)φ ′ −
(p + 2)(p + 1) 2
φ p φ ′ + p(p − 1)
φ p−2 2
3
φ′ .
Differentiating (10) with respect to ξ , we find that the right side of the above equality equals zero, that is Hc φ ′ = 0. We know that φ ′ has exactly one zero. Therefore, zero is a simple eigenvalue of Hc . By using the Sturm–Liouville theorem, we obtain that Hc only has a negative eigenvalue, whose corresponding eigenfunction is denoted by χ . By the Weyl spectral theorem, the essential spectrum of Hc belongs to [c − 2k, +∞). That completes the proof. Introducing notation I (u) = I (u; k, c ) =
(c − 2k) u2 + cu2x dx,
K ( u) = −
R
up+2 + up u2x dx.
R
Suppose that ψ satisfies the functional extremum Mλ = inf {I (u) |u ∈ X , K (u) = λ, λ > 0} . There exists a Lagrange multiplier µ such that
(2c − 4k) ψ − 2c ψ ′′ = 2µ (p + 2) ψ p+1 − pψ p−1 ψx2 − 2ψ p ψxx , 1
hence φ = µ p ψ is a solution of Eq. (10). By the homogeneity of I (u) and K (u), φ satisfies
m = m(k, c ) = inf
I ( u) 2
(K (u)) p+2 2
: u ∈ X , K ( u) > 0 .
It is easily known that Mλ = mλ p+2 , d(c ) = I (φ) = −
p+2 2
K (φ),
1 2
(I (φ) + K (φ)). Multiplying by φ both sides of (10), we have
p p d(c ) = − K (φ) = 4 4
p+2 2
p+p 2 m
.
(13)
Definition 3.5 ([24]). {ψn } is called a minimizing sequence, if for λ > 0, there exist limn→∞ (K (ψn )) = λ and limn→∞ (I (ψn )) = Mλ . Lemma 3.6. Let {ψn } be a minimizing sequence for some λ > 0, if c > 2k, then there exists a subsequence (renamed {ψn }) and scalars yn ∈ R and ψ ∈ X such that {ψn (· − yn )} → ψ , the function ψ achieves the minimum I (ψ) = Mλ subject to the constraint K (ψ) = λ. The proof is similar to that of Theorem in [24] according to the definition of I (u) and K (u). Lemma 3.7. For fixed k ∈ R, m = m(k, c ) is monotonically increasing in c.
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Proof. We assume that φc1 , φc2 are solutions of Eq. (10) corresponding to c = c1 , c = c2 , respectively, without loss of generality, let c1 < c2 , we have
2 ′ 2 dx + c φ c − 2k φ ( ) 1 1 c c 2 R I (φc2 ; k, c1 ) 2 p+2 2 = p+2 2 k φc2 k φc2 2 2 2 ′ 2 ′ 2 ′ 2 dx + φ φ dx + c + φ φ dx − c + c φ c − 2k φ ( ) c 1 c 2 2 2 c c c c 2 2 2 R R R 2 2 2 p+2 2 k φ c2 2 φc2 + φc′ 2 2 dx R m(k, c2 ) + (c1 − c2 ) p+2 2 k φc2 m(k, c2 ).
m(k, c1 ) ≤
=
= ≤
The result of Lemma 3.7 holds. The proof is completed.
It follows formula (13) and d(c ) is monotonically in c that c (u) = d−1 − 4 K (u) . A tubular neighborhood around the orbital φ(· − y) is defined by
p
Uε = u ∈ X inf ∥u − φ(· − y)∥H 2 < ε . y∈R
Lemma 3.8. If d′′ (c ) > 0, there exists ε > 0, for any u ∈ Uε , we have 1 ′′ d (c )(c (u) − c )2 . 4
E (u) − E (φ) + c (u)(Q (u) − Q (φ)) ≥
Proof. By using d′ (c ) = Q (φ) and Taylor’s formula, we yield d(c1 ) = d(c ) + Q (φ)(c1 − c ) +
1 ′′ d (c )(c1 − c )2 + o(|c1 − c |2 ) 2
(14)
where c1 near c, ε is sufficiently small for u ∈ Uε . By (14) we obtain d(c (u)) ≥ d(c ) + Q (φ)(c (u) − c ) +
1 ′′ d (c )(c (u) − c )2 4
1
= E (φ) + c (u)Q (φ) + d′′ (c )(c (u) − c )2 . 4
It follows (13) that K (φc (u) ) = −
(c (u)) = K (u), and φc (u) satisfies Mλ ,
4 d p
1
(I (u; k, c (u)) + K (u)) 2 1 ≥ I (φc (u) ; k, c (u)) + K (φc (u) ) = d(c (u)) 2
E (u) + c (u)Q (u) =
and E (u) + c (u)Q (u) ≥ E (φ) + c (u)Q (φ) + After the transposition, the result holds.
1 ′′ d (c )(c (u) − c )2 .
4
n Proof of Theorem 2.2. Suppose that φ(· − y) is unstable, there exists ε0 and initial sequence u0 satisfying infy un0 − φ(·)H 2 → 0, however,
sup inf un (t ) − φ(· − y)H 2 ≥ ε0 ,
y
t
where u is a solution of Eq. (3) with initial value un0 . Because of the continuity of un in t, we may choose some tn such that n
inf un (tn ) − φ(· − y)H 2 = ε0
y
(15)
X. Liu et al. / J. Math. Anal. Appl. 398 (2013) 776–784
781
when tn → 0, un0 − φ(·)H 2 → 0, E and Q are continuous and conservation, by Lemma 3.8 we have E (un (tn )) − E (φ) + c (un (tn ))(Q (un (tn )) − Q (φ)) ≥
1 ′′ d (c )(c (un (tn )) − c )2 . 4
(16)
It follows (15) that |un (tn )| < ∞. By (16), we know limn→∞ c (un (tn )) = c. Further,
4 n 4 lim K (u (tn )) = lim − d c u (tn ) = − d(c ) n
n→∞
p
n→∞
(17)
p
and 1 2
I (un (tn )) = E (un (tn )) + cQ (un (tn )) −
1 2
K (un (tn )) 1
= d(c ) + E (un (tn )) − E (φ) − c (Q (φ) − Q (un (tn ))) − K (un (tn )). 2
Let n → ∞, yielding lim I (un (tn )) =
2(p + 2)
n→∞
p
d(c ).
(18)
From (17) and (18), {un (tn )} is a minimizing sequence. Therefore there exists a convergent subsequence, which converges to φ . This contradicts (15). 4. Proof of Theorem 2.3 We need the following lemma to consider the proof of Theorem 2.3. Lemma 4.1. There exists ε > 0 and C 1 map α : Oε → R, for u ∈ R, r ∈ R such that: (i) u (· + α (u)) , φ ′ = 0; (ii) α(· + r ) = α(u) − r;
(iii) α ′ (u) =
φ ′ (·−α(u))
⟨u,φ ′′ (·−α(u))⟩ .
Proof. The proof is standard. See Theorem 4.1 in [23].
Introducing the new operator Bφ (u) = z (· − α (u)) −
⟨u, τ z (· − α (u))⟩ ′ J φ (· − α (u)) , ⟨u, φ ′′ (· − α (u))⟩
where u ∈ Uε , z ∈ X . Bφ (u) is an extension of formula (4.2) in [23], and a similar formula was also used in [24]. The important properties of Bφ (u) are expressed in the following auxiliary result and will be in the proof of Theorem 2.3. Lemma 4.2. On the assumption that φ ∈ L2 , which satisfy φ ′ , φ ′′ ∈ H 2 , then (i) (ii) (iii)
map Bφ : Uε → R ∈ C 1 ; Bφ (u (· + r)) = (Bφ u)(· + r ); Bφ (u) , τ u = 0, ∀u ∈ Uε .
Proof. Bφ (u) ∈ X because z ∈ X and φ ′′ ∈ H 2 . The fact that Bφ (u) is a C 1 functional follows from a straightforward explicit calculation. For more detail see Proposition 4.1 in [23] or Lemma 4.7 in [24]. By the variable substitution and (ii) of Lemma 4.1, we have
⟨u(· + r ), τ z (· − α(u) + r )⟩ ′ J φ (· − α(u) + r ) ⟨u(· + r ), φ ′′ (· − α(u) + r )⟩ ⟨u, τ z (· − α(u))⟩ ′ = z (· − α(u) + r ) − J φ (· − α(u) + r ) ⟨u, φ ′′ (· − α(u))⟩ = (Bφ u)(· + r ).
Bφ (u(· + r )) = z (· − α(u) + r ) −
So (ii) holds. From
⟨Bφ (u), τ u⟩ = ⟨z , τ u⟩ − that (iii) holds.
⟨u, τ z ⟩ ⟨u, τ z ⟩ ′ ⟨J φ , τ u⟩ = ⟨τ z , u⟩ − ⟨τ τ −1 ∂x φ ′ , u⟩ = 0 ′′ ⟨u, φ ⟩ ⟨u, φ ′′ ⟩
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X. Liu et al. / J. Math. Anal. Appl. 398 (2013) 776–784
Lemma 4.3. Suppose that u = R(κ, u) is a solution of initial problem
du
= Bφ (u), dκ u(0) = u0 .
then (i) u ∈ Uε , map R ∈ C 1 ; (ii) Q (R(κ, u)) doesn’t depend on κ . Proof. It follows Lemma 4.2 that (i) holds and dQ dκ (ii) holds.
= τ u,
du dκ
= τ u, Bφ (u) = 0,
Lemma 4.4. There exists a C 1 function Λ : {v ∈ Uε , Q (v) = Q (φ)} → R, for any v ∈ Uε , and satisfies E (R (Λ(v), v)) > E (φ). The proof is similar to Lemma 4.3 in [21] or Lemma 4.2 in [23]. Lemma 4.5. For any v ∈ Uε and Q (v) = Q (φ), then E (φ) < E (v) + Λ(v) E ′ (v), Bφ (v) .
Proof. By Taylor’s theorem and Lemma 4.4, we have
E (φ) < E (R (Λ(v), v)) = E (v) + Λ(v) E (v), ′
dR
Λ(v)
+ o (Λ(v)) ,
and E (φ) < E (v) + Λ(v) E ′ (v), Bφ (v) .
Lemma 4.6. Let be ω near c, then
⟨Hω χ , χ⟩ < 0, where χ is represented by Lemma 3.4. Proof. By Lemma 3.4, Hc has a unique simple negative eigenvalue, that is
⟨Hc χ , χ⟩ = −β 2 χ , χ = −β 2 < 0, where χ is the corresponding eigenfunction of negative eigenvalue −β 2 . By the continuity of ⟨Hc χ , χ⟩ in c, taking ε = − ⟨Hc χ2 ,χ⟩ , we have
−ε ≤ ⟨Hω χ , χ⟩ − ⟨Hc χ , χ⟩ ≤ ε, hence
⟨Hω χ , χ⟩ ≤
⟨Hc χ , χ⟩ 2
< 0.
Lemma 4.7. If d′′ (c ) < 0, there exists a C 2 curve ψω : (−δ, δ) → Uε , in which Q (ψω ) = Q (φ), the sign of E ′ (ψω ), Bφ (ψω ) vary at ω = c, and
E (ψω ) < E (φ). Proof. Introduce that P (ω, σ ) = Q (φω + σ χ ), where ω near c, then
∂ P dφω ′ = Q (φω + σ χ ), = d′′ (c ) < 0. ∂ω σ = 0 dω σ = 0 ω=c ω=c
X. Liu et al. / J. Math. Anal. Appl. 398 (2013) 776–784
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By the implicit theorem, there exists a C 1 function ω : (−δ, δ) → R, and ω(0) = c. Let C 2 be a curve denoted by ψω = φω + σ χ , in the neighborhood of (c , 0), ψω ∈ Uε and Q (φ) = Q (ψω ). By Taylor series expansion of Lω (ψω ) = E (ψω ) + ωQ (ψω ) in σ = 0, yields Lω (ψω ) = Lω (φω ) + σ L′ω (φω ), χ +
σ2 2
L′′ω (φω )χ , χ + o(σ 2 ).
According to L′ω (φω ) = 0, L′′ω (φω ) = Hω . We obtain E (φω + σ χ ) = d(ω) − ωQ (φ) +
σ2 2
⟨Hω χ , χ⟩ .
(19)
By Taylor series expansion of d(ω) in ω = c, we have d(ω) = d(c ) + d′ (c )(ω − c ) +
1 ′′ d (c )(ω − c )2 + o((ω − c )2 ). 2
From d′′ (c ) < 0, d′ (c ) = Q (φ) and the above equality, we have d(w) < d(c ) + d′ (c )(ω − c ) = E (φ) + ωQ (φ).
(20)
It follows (19), (20) and Lemma 4.6 that E (φω + σ χ ) < E (φ). Apply Lemma 4.5 with v = ψω to derive that Λ(ψω ) E ′ (ψω ), Bφ (ψω ) > 0 for w ̸= c. Therefore, to verify the sign of
E ′ (ψω ), Bφ (ψω ) vary at ω = c, we only consider the alteration of the sign of Λ(ψω ). For
⟨G(R(κ, u)) − φ, χ⟩ = 0, where G(u) = u(· + α(u)). Let u = R(Λ(ψω ), ψω ) and differentiating the above equation by ω, yields
G′ (φ)
dΛ(ψω ) dω
dψ
+1
dψω
= 0. , χ dω ω=c
Because of G′ (φ) dωω , χ ̸= 0, hence ω=c dΛ(ψω ) dω
= −1 ̸= 0
and
Λ(ψω ) = Λ(φ) − (ω − c ) + o((ω − c )).
(21)
Observing that (21), when ω through c, we find that the sign of Λ(ψω ) change. The proof is completed.
Proof of Theorem 2.3. For some u0 ∈ X and u0 near φ , but u(x, t ) run out Uε in finite time, where u(x, t ) is a solution of Eq. (3) with initial value u0 , which demonstrates φ(· − y) is unstable. Noting that [0, T ] is maximum time interval when u(x, t ) lies in Uε . In the following we will prove T < +∞. Introducing that A(t ) = R Z (x − α(u(t )))udx, where t ∈ [0, T ],
+∞
Z (x) = −∞ τ z (x)dx. By Lemma 3.3, we have
|A(t )| =
+∞
[Y (x − α(u)) − γ H (x − α(u))] udx + γ −∞
+∞
α(u)
udx
3 4
≤ ∥Y − γ H ∥L2 ∥u∥L2 + r0 (1 + t ) < +∞,
(22)
+∞
where H is the Heaviside step function, r0 depends on u0 and γ = −∞ τ z (x)dx. Therefore, dA(t ) dt
du
du
= −α (u) τ z (· − α(u))udx + Z (· − α(u)) dx dt R dt R du du ⟨τ z (· − α(u)), u⟩ + Z (· − α(u)), = − α ′ (u), dt dt du = Z (· − α(u)) − α ′ (u) ⟨τ z (· − α(u)), u⟩ , dt = Z (· − α(u)) − α ′ (u) ⟨τ z (· − α(u)), u⟩ , JE ′ (u) = − JZ (· − α(u)) − J α ′ (u) ⟨τ z (· − α(u)), u⟩ , E ′ (u) ′
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X. Liu et al. / J. Math. Anal. Appl. 398 (2013) 776–784
= − τ −1 τ z (· − α(u)) − J α ′ (u) ⟨τ z (· − α(u)), u⟩ , E ′ (u) = − Bφ (u), E ′ (u) .
In this process we take advantage of the definition of Bφ (u) and the property of J. By Lemmas 3.2 and 4.7, we know that E (φ) − E (u) = E (φ) − E (u0 ) > 0. It follows Lemma 4.5, u ∈ Uε and Λ(φ) = 0 that Λ(u)⟨E ′ (u), Bφ (u)⟩ > 0. We may choose Λ(u) < 1 and have
⟨E ′ (u), Bφ (u)⟩ ≥ E (φ) − E (u0 ) > 0. Therefore, there exists ε3 > 0, such that
−
dA(t ) dt
≥ ε3 > 0,
t ∈ [0, T ].
(23)
By (22) and (23), we have
−
A(T ) − A(0) T
≥ ε3 ,
and T ≤−
A(T ) − A(0)
ε3
≤
| 2A(t ) | < +∞, ε3
which means that u(·, t ) eventually leaves the tube Uε . This implies instability, and completes the proof of Theorem 2.3.
Acknowledgments This project is Supported by United Fund of Guizhou Science and Technology Department and Guizhou Minzu University (No. LKM[2011]14), Natural Science Fund of Guizhou Education Department (No. 2010026), Key Laboratory Construction Project ‘‘Pattern Recognition and Intelligent System’’ of Guizhou Province (No. [2009]4002) and Graduate Student Education Innovation Fund ‘‘Message Processing and Pattern Recognition’’ of Guizhou Province. Scientific and technical innovation talents support plan of Guizhou Education Department. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
R. Camassa, D. Holm, An integrable shallow water equation with peaked solitons, Physics Review Letters 71 (1993) 1661–1664. S.J. Kamdem, Z.J. Qiao, Decomposition method for the Camassa–Holm equation, Chaos, Solitons and Fractals 31 (2007) 437–447. F. Copper, H. Shepard, Solitons in the Camssa-Holm shallow water equation, Physics Letters A 194 (1994) 246–250. A. Constantin, A. Strauss, Stability of peakons, Communications on Pure and Applied Mathematics LIII (2000) 0603–0610. A. Constantin, A. Strauss, Stability of a class of solitary waves in compressible elastic rods, Physics Letters A 270 (2000) 140–148. H. Mckean, The Liouville correspondence between the KdV and the Camassa–Holm equation, Communications on Pure and Applied Mathematics LVI (2003) 0998–1015. J.P. Boyd, Peakons and coshoidal waves: travelling wave solutions of the Camassa–Holm equation, Applied Mathematics and Computation 81 (1997) 173–187. Z. Liu, R. Wang, Z. Jing, Peaked wave solutions of Camassa–Holm equation, Chaos, Solitons and Fractals 19 (2004) 77–92. A. Constantin, W.A. Strauss, Stability of the Camassa–Holm Solitons, Journal of Nonlinear Science 12 (2002) 415–422. A. Constantin, L. Molinet, Orbital stability of solitary waves for a shallow water equation, Physica D 157 (2001) 75–89. L.X. Tian, J.L. Yin, New compacton solitons and solitary wave solutions of fully nonlinear generalized Camassa–Holm equations, Chaos Solitons and Fractals 20 (2004) 289–299. A. Wazwaz, A class of nonlinear fourth order variant of a generalized Camassa–Holm equation with compact and noncompact solutions, Applied Mathematics and Computation 165 (2005) 485–501. A. Wazwaz, New compact and noncompact solutions for two variants of a modified Camassa–Holm equation, Applied Mathematics and Computation 163 (2005) 1165–1179. R.A. Kraenkel, A. Zenchuk, Two dimensional integrable generalization of the Camassa–Holm equation, Physics Letters A 260 (1999) 218–224. Z. Liu, T. Qian, M. Tang, Peakons and their bifurcation in a generalized Camassa–Holm equation, International Journal of Bifurcation and Chaos 11 (2002) 781–792. T. Qian, M. Tang, Peakons and periodic cusp waves in a generalized Camassa–Holm equation, Chaos, Solitons and Fractals 12 (2001) 1347–1360. Z. Liu, T. Qian, Peakons of the Camassa–Holm equation, Applied Mathematical Modelling 26 (2002) 473–480. T. Tian, X. Song, New peaked solitary wave solutions of the generalized Camassa–Holm equation, Chaos, Solitons and Fractals 19 (2004) 621–637. S.A. Khuri, New ansätz for obtaining wave solutions of the generalized Camassa–Holm equation, Chaos, Solitons and Fractals 25 (2005) 705–710. S. Hakkaev, K. Kirchev, Local well-posedness and orbital stability of solitary wave solutions for the generalized Camassa–Holm equation, Communications in Partial Differential Equation 30 (2005) 761–781. M. Grillakis, J. Shatah, W. Strauss, Stability theory of solitary waves in the presence of symmetry I, Journal of Functional Analysis 74 (1987) 160–197. S. Hakkaev, K. Kirchev, On the well-posedness and stability of peakons for the generalized Camassa–Holm equation, International Journal of Nonlinear Science 1 (2006) 139–148. J.L. Bona, P.E. Souganidis, W.A. Strauss, Stability and instability of solitary waves of KdV type, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 411 (1987) 395–412. L. Steve, Y. Liu, Stability of solitary waves of a generalized Ostrovsky equation, SIAM Journal on Mathematical Analysis 38 (2006) 985–1011.
Contents lists available at SciVerse ScienceDirect
Journal of Mathematical Analysis and Applications journal homepage: www.elsevier.com/locate/jmaa
The orbital stability of the solitary wave solutions of the generalized Camassa–Holm equation✩ Xiaohua Liu a , Weiguo Zhang b,∗ , Zhengming Li b a
Guizhou Minzu University, GuiYang 550025, PR China
b
University of Shanghai for Science and Technology, ShangHai 200093, PR China
article
info
Article history: Received 27 April 2010 Available online 23 September 2012 Submitted by Thomas P. Witelski Keywords: Generalized Camassa–Holm equation Stability Solitary wave solution
abstract In this paper, we consider the orbital stability of smooth solitary wave solutions of the generalized Camassa–Holm equation. By constructing the functional extremum problem and using the orbital stability theory presented by Grillakis, Shatah, Strauss and Bona, and Souganidis, we show that the solitary wave solutions of the generalized Camassa–Holm equation are orbitally stable or unstable as determined by the sign of a discriminant. The conclusions presented by the previous authors, such as Hakkaev and Kirchev, Constantin and Strauss, can be considered as a special case of our results. © 2012 Elsevier Inc. All rights reserved.
1. Introduction In 1993, Camassa and Holm [1] derived the shallow water wave equation ut + 2kux − uxxt + 3uux = 2ux uxx + uuxxx .
(1)
Eq. (1) is called the Camassa–Holm (CH) equation, which models the unidirectional propagation of water waves in the shallow water regime, when the wavelength is considerably larger than the average water depth. Where u denotes the fluid velocity, or can also be interpreted as the height of the water’s free surface above a flat bottom, and the constant k is related to the critical shallow water wave speed. For the special case k = 0, they [1] showed that (1) has the peaked solitary waves u(x, t ) = c exp(−|x − ct |), which were called peakons due to the discontinuity of the first derivation at the wave vertex; By applying the decomposition method to Eq. (1), three approximate solutions were obtained [2], which are similar to the peaked solitary waves. The CH equation has been investigated [3–10] extensively. Boyd [7] investigated the peaked periodic waves that have a discontinuous first derivative at each peak called coshoidal waves or periodic cusp waves. The analytic expressions of peaked solitary waves and peaked periodic waves of CH Eq. (1) were obtained in [8] by using the bifurcation method of planar dynamical systems. The orbital stability of smooth solitary wave solutions of Eq. (1) is considered by Constantin and Strauss [9], and the stability of the non-smooth peakons (k = 0) by Constantin [4] and Constantin [10] by means of variational methods.
✩ This project is supported by National Natural Science Foundation of China (No. 11071164), Scientific Research Innovation Project of Shanghai Municipal Education Commission (No. 13ZZ118), Shanghai Natural Science Foundation (No. 10ZR1420800) and Leading Academic Discipline Project of Shanghai Municipal Government. ∗ Corresponding author. E-mail address: [email protected] (X. Liu).
0022-247X/$ – see front matter © 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2012.09.014
X. Liu et al. / J. Math. Anal. Appl. 398 (2013) 776–784
777
If the nonlinear convection term uux of Eq. (1) is changed into the new nonlinear convection term um ux , we have the following modified Camassa–Holm(MCH) equation ut + 2kux − uxxt + aum ux = 2ux uxx + uuxxx .
(2)
Because of the new nonlinear convection term, the Eq. (2) creates many new nonlinear phenomena, such as compacton solitons with compact support, solitons with cusps, or peakons [11–18]. Tian and Yin [11] proposed four simple ansätzs to obtain abundant solutions: compactons, solitary patterns solutions having infinite slopes or cusps, and solitary waves. Qian and Tang [16] studied the peakons and the periodic cusp wave solution of the MCH Eq. (2), they constructed some smooth periodic wave solutions, periodic cusp wave solutions, and oscillatory solitary wave solutions. Liu and Qian [17] investigated the peakons and their bifurcation of the MCH Eq. (2), and for special cases they gave some explicit expressions for the peakons. Tian and Song [18] derived some new exact peaked solitary waves. Khuri [19] studied the periodic wave and peaked solitary waves of Eq. (2) by employing polynomial ansätz. Considered herein is the stability of the solitary wave solutions to the generalized Camassa–Holm (GCH) equation
ut − uxxt +
2ku +
p+2 2
u
p+1
= pu
2 p−1 ux
2
x
p
,
+ u uxx
(3)
x
where k ∈ R, p > 0. The GCH Eq. (3) reduces to the CH Eq. (1) for p = 1. We know that the Eq. (3) has more new nonlinear phenomena and structures than the Eqs. (1) and (2). For any p, the explicit or implicit expression of traveling wave solutions to Eq. (3) will have become more and more complex, but so far have not been presented in papers. Solitary waves are valuable in theory and practice, in applications when they are stable in some sense. It is significant to study the orbital stability of solitary waves. There exist a few papers on the stability of solitary waves of Eq. (3). Hakkev [20] establish the orbital stability of smooth solitary wave solutions of Eq. (3) by virtue of the orbital stability theory presented by Grillakis–Shatah–Strauss [21], and the local well posedness of the Cauchy problem of Eq. (3) in the Sobolev space H s (s > 23 ) by applying the method of the pseudoparabolic regularization. Hakkev [22] discussed the stability of non-smooth peakons of Eq. (3) by the method employed by Constantin [4], and the well posedness by using Kato’s semigroup approach. In this paper, the main results are represented in Section 2. In Section 3, the stability of solitary wave solutions of Eq. (3) is studied with the help of the orbital stability theory [21]. In Section 4, the instability of solitary waves of Eq. (3) is considered by using the theory reported by [23]. Then we can deduce the results of [20] using the analysis done here. Specially, by using the method of the third and fourth section, we can establish the proof of Theorem 4.4 in [20]. 2. Main results and notation Notation: Let Ls = Ls (R) be the Lebesgue measurable space with
|f (x)| dx s
∥f ∥Ls =
1s
,
R
and H s = H s (R) be the Sobolev space with
2 (1 + |ξ |2 )s gˆ (ξ ) dξ
∥g ∥H s =
21
,
R
where gˆ (ξ ) = R e−iξ x g (x)dx. For any Banach space X and C ([0, T ], X ) is the set of all continuous functions from [0, T ] to X . Eq. (3) can be rewritten in a Hamiltonian form
du dt
= JE ′ (u),
u ∈ X,
(4)
where X = H 2 , whose dual space is denoted by X ∗ = H −2 , τ = 1 − ∂x2 . J = τ −1 ∂x is a skew-symmetric linear operator. The Hamiltonian is E (u) = −
2
ku +
up+2
R
2
+
up 2
u2x
dx.
(5)
It follows (5) that E ′ (u) = −2ku − E ′′ (u) = −2k −
p+2 2
up+1 +
p 2
(p + 2)(p + 1) 2
up−1 u2x + up uxx ,
up +
Let the inner product of X be (u, v) =
p(p − 1) 2
up−2 u2x + pup−1 uxx + pup−1 ux ∂x + up ∂x2 .
(6) (7)
(uv + ux vx + uxx vxx )dx. For any u, v ∈ X , there exists a natural isomorphism 4 2 I : X → X defined by⟨Iu, v⟩ = (u, v), where ⟨u, v⟩ = R uv dx, I = ∂∂x4 − ∂∂x2 + 1. Let T be a one parameter group of unitary ∗
R
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X. Liu et al. / J. Math. Anal. Appl. 398 (2013) 776–784
operator on X defined by T (s)u(·) = u(· − s), s ∈ R, u ∈ X . So T ′ (0) = − ∂∂ and JB = T ′ (0), then B = −(1 − ∂x2 ). Defining x the functional 1 1 Q (u) = − ⟨Bu, u⟩ = − 2 2
(u2 + u2x )dx.
(8)
Q ′′ (u) = 1 − ∂x2 .
(9)
R
It follows (8) that Q ′ (u) = −Bu = u − uxx ,
Defining orbital φ(· − y) = {T (y)φ(·)|y ∈ R}. They may be interpreted physically as ‘‘solitary waves’’ or ‘‘bound states’’. Definition 2.1. The orbital φ(· − y) is stable if for any ε > 0, there exists δ > 0 with the following property. If ∥u(·, 0) − φ(·)∥H 2 < δ , for some T > 0, u ∈ C ([0, T ], H 2 ), and u is a solution of Eq. (3) with u(·, 0) = u0 , then u can be continued to a solution in 0 ≤ T < ∞ and sup inf ∥u (·, t ) − φ (· − y)∥H 2 < ε. t
y
Otherwise, the orbital φ(· − y) is called unstable. Definition 2.1 can be seen in Refs. [20,21,23]. Substituting u(x, t ) = φ(x − ct ) = φ(ξ ) into Eq. (3), integrating once with respect to ξ , taking integral constant zero, Eq. (3) becomes c φ − c φ ′′ − 2kφ −
p+2 2
φ p+1 + pφ p−1
φ′2 2
+ φ p φ ′′ = 0.
(10)
By (6) and (9), Eq. (10) can be rewritten as E ′ (φ) + cQ ′ (φ) = 0, φ is called a bound-state solution. To study the orbital stability of the solitary wave solutions of Eq. (3), we need operator Hc = E ′′ (φ) + cQ ′′ (φ) and function d(c ) = E (φ) + cQ (φ). From (7) and (9), we have Hc = ∂x (φ p − c )∂x + c − 2k −
(p + 2)(p + 1) 2
φp +
p(p − 1) 2
2
φ p−2 φ ′ + pφ p−1 φ ′′ .
(11)
First, we give the main results: Theorem 2.2. Suppose that φ(· − y) is a solitary wave solution, for c > 2k, if d′′ (c ) > 0, then φ(· − y) is stable. Theorem 2.3. Suppose that φ(· − y) is a solitary wave solution, for c > 2k, if d′′ (c ) < 0, then φ(· − y) is unstable. Remark 2.4. By applying Theorems 2.2 and 2.3 to the smooth solitary wave solutions to Eq. (3), we can yield Theorem 4.4 and Theorem 4.5 in [20] and Theorem in [9](p = 1). 3. Proof of Theorem 2.2 To give the proof of Theorem 2.2, we need the following lemma: Lemma 3.1. Suppose that u0 ∈ H s , s > 23 , Then there is a real number T > 0 and a unique solution u to the initial-value problem of Eq. (3) such that u ∈ C ([0, T ), H s ) satisfying u(x, 0) = u0 . Proof. Proof see Theorem 3.4 in [20].
Lemma 3.2. Let u be a solution of Eq. (3) satisfying u(x, 0) = u0 , then E (u) = E (u0 ), Q (u) = Q (u0 ), and I (u) = I (u0 ) =
u0 (x)dx
R
convergence. Proof. Integrating Eq. (4) over x ∈ [a, b], t ∈ [0, t ], we have
t
b
t
b
ut dxdt = 0
a
0
p + 2 p+1 p τ −1 ∂x −2ku − u + up−1 u2x + up uxx dxdt .
a
2
2
Letting a → +∞, b → −∞, it follows that u and its derivation decay to zero at infinity such that I (u) = I (u0 ) =
u0 (x)dx
R
converge. For proofs of the first and second equalities, we refer the readers to Ref. [20].
X. Liu et al. / J. Math. Anal. Appl. 398 (2013) 776–784
Lemma 3.3. Let u0 ∈ H s (s >
sup −∞
z
−∞
779
), and u(x, t ) is a solution of Eq. (3) satisfying u(x, 0) = u0 . Then 3 u(x, t )dx ≤ r0 (1 + t ) 4 , 3 2
(12)
where r0 depends on u0 . Proof. See Theorem 4.1 of [20] or Theorem 2.2 of [23]. Lemma 3.4. For each c ∈ (2k, ∞), Hc has a unique simple negative eigenvalue, zero is a simple eigenvalue, and the rest of its spectrum is bounded away from zero. Proof. For any u, v ∈ X , we have
φ uxx v dx = p
R
pφ
p−1
R
uxx v dx =
φx vx udx +
R
R
uvxx dx and
φ vxx udx − p
R
pφ p−1 φx ux v dx. R
Therefore,
p ∂x φ − c ∂x u, v = ∂x φ p − c ∂x v, u . By (11), we know that Hc is a self-conjugate operator and Hc φ ′ = (φ p − c )φ ′′′ + 2pφ p−1 φ ′ φ ′′ + (c − 2k)φ ′ −
(p + 2)(p + 1) 2
φ p φ ′ + p(p − 1)
φ p−2 2
3
φ′ .
Differentiating (10) with respect to ξ , we find that the right side of the above equality equals zero, that is Hc φ ′ = 0. We know that φ ′ has exactly one zero. Therefore, zero is a simple eigenvalue of Hc . By using the Sturm–Liouville theorem, we obtain that Hc only has a negative eigenvalue, whose corresponding eigenfunction is denoted by χ . By the Weyl spectral theorem, the essential spectrum of Hc belongs to [c − 2k, +∞). That completes the proof. Introducing notation I (u) = I (u; k, c ) =
(c − 2k) u2 + cu2x dx,
K ( u) = −
R
up+2 + up u2x dx.
R
Suppose that ψ satisfies the functional extremum Mλ = inf {I (u) |u ∈ X , K (u) = λ, λ > 0} . There exists a Lagrange multiplier µ such that
(2c − 4k) ψ − 2c ψ ′′ = 2µ (p + 2) ψ p+1 − pψ p−1 ψx2 − 2ψ p ψxx , 1
hence φ = µ p ψ is a solution of Eq. (10). By the homogeneity of I (u) and K (u), φ satisfies
m = m(k, c ) = inf
I ( u) 2
(K (u)) p+2 2
: u ∈ X , K ( u) > 0 .
It is easily known that Mλ = mλ p+2 , d(c ) = I (φ) = −
p+2 2
K (φ),
1 2
(I (φ) + K (φ)). Multiplying by φ both sides of (10), we have
p p d(c ) = − K (φ) = 4 4
p+2 2
p+p 2 m
.
(13)
Definition 3.5 ([24]). {ψn } is called a minimizing sequence, if for λ > 0, there exist limn→∞ (K (ψn )) = λ and limn→∞ (I (ψn )) = Mλ . Lemma 3.6. Let {ψn } be a minimizing sequence for some λ > 0, if c > 2k, then there exists a subsequence (renamed {ψn }) and scalars yn ∈ R and ψ ∈ X such that {ψn (· − yn )} → ψ , the function ψ achieves the minimum I (ψ) = Mλ subject to the constraint K (ψ) = λ. The proof is similar to that of Theorem in [24] according to the definition of I (u) and K (u). Lemma 3.7. For fixed k ∈ R, m = m(k, c ) is monotonically increasing in c.
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X. Liu et al. / J. Math. Anal. Appl. 398 (2013) 776–784
Proof. We assume that φc1 , φc2 are solutions of Eq. (10) corresponding to c = c1 , c = c2 , respectively, without loss of generality, let c1 < c2 , we have
2 ′ 2 dx + c φ c − 2k φ ( ) 1 1 c c 2 R I (φc2 ; k, c1 ) 2 p+2 2 = p+2 2 k φc2 k φc2 2 2 2 ′ 2 ′ 2 ′ 2 dx + φ φ dx + c + φ φ dx − c + c φ c − 2k φ ( ) c 1 c 2 2 2 c c c c 2 2 2 R R R 2 2 2 p+2 2 k φ c2 2 φc2 + φc′ 2 2 dx R m(k, c2 ) + (c1 − c2 ) p+2 2 k φc2 m(k, c2 ).
m(k, c1 ) ≤
=
= ≤
The result of Lemma 3.7 holds. The proof is completed.
It follows formula (13) and d(c ) is monotonically in c that c (u) = d−1 − 4 K (u) . A tubular neighborhood around the orbital φ(· − y) is defined by
p
Uε = u ∈ X inf ∥u − φ(· − y)∥H 2 < ε . y∈R
Lemma 3.8. If d′′ (c ) > 0, there exists ε > 0, for any u ∈ Uε , we have 1 ′′ d (c )(c (u) − c )2 . 4
E (u) − E (φ) + c (u)(Q (u) − Q (φ)) ≥
Proof. By using d′ (c ) = Q (φ) and Taylor’s formula, we yield d(c1 ) = d(c ) + Q (φ)(c1 − c ) +
1 ′′ d (c )(c1 − c )2 + o(|c1 − c |2 ) 2
(14)
where c1 near c, ε is sufficiently small for u ∈ Uε . By (14) we obtain d(c (u)) ≥ d(c ) + Q (φ)(c (u) − c ) +
1 ′′ d (c )(c (u) − c )2 4
1
= E (φ) + c (u)Q (φ) + d′′ (c )(c (u) − c )2 . 4
It follows (13) that K (φc (u) ) = −
(c (u)) = K (u), and φc (u) satisfies Mλ ,
4 d p
1
(I (u; k, c (u)) + K (u)) 2 1 ≥ I (φc (u) ; k, c (u)) + K (φc (u) ) = d(c (u)) 2
E (u) + c (u)Q (u) =
and E (u) + c (u)Q (u) ≥ E (φ) + c (u)Q (φ) + After the transposition, the result holds.
1 ′′ d (c )(c (u) − c )2 .
4
n Proof of Theorem 2.2. Suppose that φ(· − y) is unstable, there exists ε0 and initial sequence u0 satisfying infy un0 − φ(·)H 2 → 0, however,
sup inf un (t ) − φ(· − y)H 2 ≥ ε0 ,
y
t
where u is a solution of Eq. (3) with initial value un0 . Because of the continuity of un in t, we may choose some tn such that n
inf un (tn ) − φ(· − y)H 2 = ε0
y
(15)
X. Liu et al. / J. Math. Anal. Appl. 398 (2013) 776–784
781
when tn → 0, un0 − φ(·)H 2 → 0, E and Q are continuous and conservation, by Lemma 3.8 we have E (un (tn )) − E (φ) + c (un (tn ))(Q (un (tn )) − Q (φ)) ≥
1 ′′ d (c )(c (un (tn )) − c )2 . 4
(16)
It follows (15) that |un (tn )| < ∞. By (16), we know limn→∞ c (un (tn )) = c. Further,
4 n 4 lim K (u (tn )) = lim − d c u (tn ) = − d(c ) n
n→∞
p
n→∞
(17)
p
and 1 2
I (un (tn )) = E (un (tn )) + cQ (un (tn )) −
1 2
K (un (tn )) 1
= d(c ) + E (un (tn )) − E (φ) − c (Q (φ) − Q (un (tn ))) − K (un (tn )). 2
Let n → ∞, yielding lim I (un (tn )) =
2(p + 2)
n→∞
p
d(c ).
(18)
From (17) and (18), {un (tn )} is a minimizing sequence. Therefore there exists a convergent subsequence, which converges to φ . This contradicts (15). 4. Proof of Theorem 2.3 We need the following lemma to consider the proof of Theorem 2.3. Lemma 4.1. There exists ε > 0 and C 1 map α : Oε → R, for u ∈ R, r ∈ R such that: (i) u (· + α (u)) , φ ′ = 0; (ii) α(· + r ) = α(u) − r;
(iii) α ′ (u) =
φ ′ (·−α(u))
⟨u,φ ′′ (·−α(u))⟩ .
Proof. The proof is standard. See Theorem 4.1 in [23].
Introducing the new operator Bφ (u) = z (· − α (u)) −
⟨u, τ z (· − α (u))⟩ ′ J φ (· − α (u)) , ⟨u, φ ′′ (· − α (u))⟩
where u ∈ Uε , z ∈ X . Bφ (u) is an extension of formula (4.2) in [23], and a similar formula was also used in [24]. The important properties of Bφ (u) are expressed in the following auxiliary result and will be in the proof of Theorem 2.3. Lemma 4.2. On the assumption that φ ∈ L2 , which satisfy φ ′ , φ ′′ ∈ H 2 , then (i) (ii) (iii)
map Bφ : Uε → R ∈ C 1 ; Bφ (u (· + r)) = (Bφ u)(· + r ); Bφ (u) , τ u = 0, ∀u ∈ Uε .
Proof. Bφ (u) ∈ X because z ∈ X and φ ′′ ∈ H 2 . The fact that Bφ (u) is a C 1 functional follows from a straightforward explicit calculation. For more detail see Proposition 4.1 in [23] or Lemma 4.7 in [24]. By the variable substitution and (ii) of Lemma 4.1, we have
⟨u(· + r ), τ z (· − α(u) + r )⟩ ′ J φ (· − α(u) + r ) ⟨u(· + r ), φ ′′ (· − α(u) + r )⟩ ⟨u, τ z (· − α(u))⟩ ′ = z (· − α(u) + r ) − J φ (· − α(u) + r ) ⟨u, φ ′′ (· − α(u))⟩ = (Bφ u)(· + r ).
Bφ (u(· + r )) = z (· − α(u) + r ) −
So (ii) holds. From
⟨Bφ (u), τ u⟩ = ⟨z , τ u⟩ − that (iii) holds.
⟨u, τ z ⟩ ⟨u, τ z ⟩ ′ ⟨J φ , τ u⟩ = ⟨τ z , u⟩ − ⟨τ τ −1 ∂x φ ′ , u⟩ = 0 ′′ ⟨u, φ ⟩ ⟨u, φ ′′ ⟩
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Lemma 4.3. Suppose that u = R(κ, u) is a solution of initial problem
du
= Bφ (u), dκ u(0) = u0 .
then (i) u ∈ Uε , map R ∈ C 1 ; (ii) Q (R(κ, u)) doesn’t depend on κ . Proof. It follows Lemma 4.2 that (i) holds and dQ dκ (ii) holds.
= τ u,
du dκ
= τ u, Bφ (u) = 0,
Lemma 4.4. There exists a C 1 function Λ : {v ∈ Uε , Q (v) = Q (φ)} → R, for any v ∈ Uε , and satisfies E (R (Λ(v), v)) > E (φ). The proof is similar to Lemma 4.3 in [21] or Lemma 4.2 in [23]. Lemma 4.5. For any v ∈ Uε and Q (v) = Q (φ), then E (φ) < E (v) + Λ(v) E ′ (v), Bφ (v) .
Proof. By Taylor’s theorem and Lemma 4.4, we have
E (φ) < E (R (Λ(v), v)) = E (v) + Λ(v) E (v), ′
dR
Λ(v)
+ o (Λ(v)) ,
and E (φ) < E (v) + Λ(v) E ′ (v), Bφ (v) .
Lemma 4.6. Let be ω near c, then
⟨Hω χ , χ⟩ < 0, where χ is represented by Lemma 3.4. Proof. By Lemma 3.4, Hc has a unique simple negative eigenvalue, that is
⟨Hc χ , χ⟩ = −β 2 χ , χ = −β 2 < 0, where χ is the corresponding eigenfunction of negative eigenvalue −β 2 . By the continuity of ⟨Hc χ , χ⟩ in c, taking ε = − ⟨Hc χ2 ,χ⟩ , we have
−ε ≤ ⟨Hω χ , χ⟩ − ⟨Hc χ , χ⟩ ≤ ε, hence
⟨Hω χ , χ⟩ ≤
⟨Hc χ , χ⟩ 2
< 0.
Lemma 4.7. If d′′ (c ) < 0, there exists a C 2 curve ψω : (−δ, δ) → Uε , in which Q (ψω ) = Q (φ), the sign of E ′ (ψω ), Bφ (ψω ) vary at ω = c, and
E (ψω ) < E (φ). Proof. Introduce that P (ω, σ ) = Q (φω + σ χ ), where ω near c, then
∂ P dφω ′ = Q (φω + σ χ ), = d′′ (c ) < 0. ∂ω σ = 0 dω σ = 0 ω=c ω=c
X. Liu et al. / J. Math. Anal. Appl. 398 (2013) 776–784
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By the implicit theorem, there exists a C 1 function ω : (−δ, δ) → R, and ω(0) = c. Let C 2 be a curve denoted by ψω = φω + σ χ , in the neighborhood of (c , 0), ψω ∈ Uε and Q (φ) = Q (ψω ). By Taylor series expansion of Lω (ψω ) = E (ψω ) + ωQ (ψω ) in σ = 0, yields Lω (ψω ) = Lω (φω ) + σ L′ω (φω ), χ +
σ2 2
L′′ω (φω )χ , χ + o(σ 2 ).
According to L′ω (φω ) = 0, L′′ω (φω ) = Hω . We obtain E (φω + σ χ ) = d(ω) − ωQ (φ) +
σ2 2
⟨Hω χ , χ⟩ .
(19)
By Taylor series expansion of d(ω) in ω = c, we have d(ω) = d(c ) + d′ (c )(ω − c ) +
1 ′′ d (c )(ω − c )2 + o((ω − c )2 ). 2
From d′′ (c ) < 0, d′ (c ) = Q (φ) and the above equality, we have d(w) < d(c ) + d′ (c )(ω − c ) = E (φ) + ωQ (φ).
(20)
It follows (19), (20) and Lemma 4.6 that E (φω + σ χ ) < E (φ). Apply Lemma 4.5 with v = ψω to derive that Λ(ψω ) E ′ (ψω ), Bφ (ψω ) > 0 for w ̸= c. Therefore, to verify the sign of
E ′ (ψω ), Bφ (ψω ) vary at ω = c, we only consider the alteration of the sign of Λ(ψω ). For
⟨G(R(κ, u)) − φ, χ⟩ = 0, where G(u) = u(· + α(u)). Let u = R(Λ(ψω ), ψω ) and differentiating the above equation by ω, yields
G′ (φ)
dΛ(ψω ) dω
dψ
+1
dψω
= 0. , χ dω ω=c
Because of G′ (φ) dωω , χ ̸= 0, hence ω=c dΛ(ψω ) dω
= −1 ̸= 0
and
Λ(ψω ) = Λ(φ) − (ω − c ) + o((ω − c )).
(21)
Observing that (21), when ω through c, we find that the sign of Λ(ψω ) change. The proof is completed.
Proof of Theorem 2.3. For some u0 ∈ X and u0 near φ , but u(x, t ) run out Uε in finite time, where u(x, t ) is a solution of Eq. (3) with initial value u0 , which demonstrates φ(· − y) is unstable. Noting that [0, T ] is maximum time interval when u(x, t ) lies in Uε . In the following we will prove T < +∞. Introducing that A(t ) = R Z (x − α(u(t )))udx, where t ∈ [0, T ],
+∞
Z (x) = −∞ τ z (x)dx. By Lemma 3.3, we have
|A(t )| =
+∞
[Y (x − α(u)) − γ H (x − α(u))] udx + γ −∞
+∞
α(u)
udx
3 4
≤ ∥Y − γ H ∥L2 ∥u∥L2 + r0 (1 + t ) < +∞,
(22)
+∞
where H is the Heaviside step function, r0 depends on u0 and γ = −∞ τ z (x)dx. Therefore, dA(t ) dt
du
du
= −α (u) τ z (· − α(u))udx + Z (· − α(u)) dx dt R dt R du du ⟨τ z (· − α(u)), u⟩ + Z (· − α(u)), = − α ′ (u), dt dt du = Z (· − α(u)) − α ′ (u) ⟨τ z (· − α(u)), u⟩ , dt = Z (· − α(u)) − α ′ (u) ⟨τ z (· − α(u)), u⟩ , JE ′ (u) = − JZ (· − α(u)) − J α ′ (u) ⟨τ z (· − α(u)), u⟩ , E ′ (u) ′
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= − τ −1 τ z (· − α(u)) − J α ′ (u) ⟨τ z (· − α(u)), u⟩ , E ′ (u) = − Bφ (u), E ′ (u) .
In this process we take advantage of the definition of Bφ (u) and the property of J. By Lemmas 3.2 and 4.7, we know that E (φ) − E (u) = E (φ) − E (u0 ) > 0. It follows Lemma 4.5, u ∈ Uε and Λ(φ) = 0 that Λ(u)⟨E ′ (u), Bφ (u)⟩ > 0. We may choose Λ(u) < 1 and have
⟨E ′ (u), Bφ (u)⟩ ≥ E (φ) − E (u0 ) > 0. Therefore, there exists ε3 > 0, such that
−
dA(t ) dt
≥ ε3 > 0,
t ∈ [0, T ].
(23)
By (22) and (23), we have
−
A(T ) − A(0) T
≥ ε3 ,
and T ≤−
A(T ) − A(0)
ε3
≤
| 2A(t ) | < +∞, ε3
which means that u(·, t ) eventually leaves the tube Uε . This implies instability, and completes the proof of Theorem 2.3.
Acknowledgments This project is Supported by United Fund of Guizhou Science and Technology Department and Guizhou Minzu University (No. LKM[2011]14), Natural Science Fund of Guizhou Education Department (No. 2010026), Key Laboratory Construction Project ‘‘Pattern Recognition and Intelligent System’’ of Guizhou Province (No. [2009]4002) and Graduate Student Education Innovation Fund ‘‘Message Processing and Pattern Recognition’’ of Guizhou Province. Scientific and technical innovation talents support plan of Guizhou Education Department. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
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