# Global Existence and Asymptotic Behavior of Solutions to the Generalized Damped Boussinesq Equation

Authors: Yinxia Wang; Hengjun Zhao
Journal: Advances in Mathematical Physics
 (2013)

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## Abstract

We investigate the Cauchy problem for the generalized damped Boussinesq equation. Under small condition on the initial value, we prove the global existence and optimal decay estimate of solutions for all space dimensionsn≥1. Moreover, when n≥2, we show that the solution can be approximated by the linear solution as time tends to infinity.


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## Body

## 1. Introduction

We investigate the Cauchy problem of the following generalized damped Boussinesq equation:(1)utt-Δu-2bΔut+αΔ2u=Δf(u)
with the initial value
(2)t=0:u=u0(x),ut=u1(x).
Here u=u(x,t) is the unknown function of x=(x1,…,xn)∈ℝn and t>0, b>0, and α>0 are constants. The nonlinear term f(u) is a given smooth function of u∈ℝ satisfying f(u)=O(u2) for u→0.It is well known that the classical Boussinesq equation(3)utt+αΔ2u-Δu=βΔ(u2)
was derived by Boussinesq [1] in 1872 to describe shallow water waves, where u(x,t) is an elevation of the free surface of fluid and the constant coefficients α and β depend on the depth of fluid and the characteristic speed of long waves. It is interesting to note that this equation also governs nonlinear string oscillations.Taking into account dispersion and nonlinearity, but in real processes viscosity also plays an important role. Varlamov considered the following damped Boussinesq equation (see [2–4]):
(4)utt-Δu-2bΔut+αΔ2u=βΔ(u2),
where b>0 and α>0 are constants. Under small condition on the initial value, Varlamov [2] obtained a classical solution to the problem (4), (2) by means of the application of both the spectral and perturbation theories. Moreover, large time asymptotics of this solution was also discussed. For the problem (4), (2) in one, two, and three space dimensions, existence and uniqueness of local solution are proved by Varlamov [3]. The author also showed that for discontinuous initial perturbations this solution is infinitely differentiable with respect to time t and space coordinates for t>0 on a bounded time interval. Existence and uniqueness of the classical solution for the problem (4), (2) in two space dimensions was proved, and the solution was constructed in the form of a series. The major term of its long-time asymptotics is calculated explicitly, and a uniform in space estimate of the residual term was given (see [4]).The main purpose of this paper is to establish the following optimal decay estimate of solutions to (1), (2) for n≥1:
(5)∥∂xku(t)∥Hs+2-k≤CE0(1+t)-n/4-k/2
for 0≤k≤s+2 and s≥max{0,[n/2]-1}. Here E0≔∥u0∥L1+∥u1∥W˙-1,1+∥u0∥Hs+2+∥u1∥Hs is assumed to be small. Moreover, when n≥2, we show that our solution u can be approximated by the solution uL to the linearized problem, namely, the problem (1), (2) with f(u)≡0. More precisely, when n≥2, we show that
(6)∥∂xk(u-uL)(t)∥Hs+2-k≤CE02(1+t)-n/4-k/2η(t)
for 0≤k≤s+2 and s≥max{0,[n/2]-1}, where η(t)=(1+t)-1/2log(2+t) for n=2 and η(t)=(1+t)-1/2 for n≥3.The study of the global existence and asymptotic behavior of solutions to hyperbolic-type equations has a long history. We refer to [5, 6] for hyperbolic equations, [7–10] for damped wave equation and [11–17] for various aspects of dissipation of the plate equation.The paper is organized as follows. In Section2, we derive the solution formula of our semilinear problem. We study the decay property of the solution operators appearing in the solution formula in Section 3. Then, in Section 4, we prove the global existence and asymptotic decay of solutions. Finally, we derive a simpler asymptotic profile which gives the approximation to the linear solution in Section 5.Notations. We give some notations which are used in this paper. Let ℱ[u] denote the Fourier transform of u defined by
(7)u^(ξ)=ℱ[u]=∫ℝn‍e-iξ·xu(x)dx,ℱ
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1For1≤p≤∞, Lp=Lp(ℝn) denotes the usual Lebesgue space with the norm ∥·∥Lp. The usual Sobolev space of s is defined by Ws,p=(I-Δ)-s/2Lp with the norm ∥f∥Ws,p=∥(I-Δ)s/2f∥Lp; the homogeneous Sobolev space of s is defined by W˙s,p=(-Δ)-s/2Lp with the norm ∥f∥W˙s,p=∥(-Δ)s/2f∥Lp; especially Hs=Ws,2, H˙s=W˙s,2. Moreover, we know that Ws,p=Lp⋂‍W˙s,p for s≥0.Finally, in this paper, we denote every positive constant by the same symbolC or c without confusion. [·] is the Gauss symbol.


## 2. Solution Formula

The aim of this section is to derive the solution formula for the problem (1), (2). We first investigate the linearized equation of (1):
(8)utt-Δu-2bΔut+αΔ2u=0
with the initial data in (2). We apply the Fourier transform to (8). This yields
(9)u^tt+2b|ξ|2u^t+(|ξ|2+α|ξ|4)u^=0.
The corresponding initial value is given as
(10)t=0:u^=u^0(ξ),u^t=u^1(ξ).
The characteristic equation of (9) is
(11)λ2+2b|ξ|2λ+|ξ|2+α|ξ|4=0.
Let λ=λ±(ξ) be the corresponding eigenvalues of (11), and we obtain
(12)λ±(ξ)=-b|ξ|2±|ξ|(b2-α)|ξ|2-1.
The solution to the problem (9), (10) in the Fourier space is then given explicitly in the form
(13)u^(ξ,t)=G^(ξ,t)u^1(ξ)+H^(ξ,t)u^0(ξ),
where
(14)G^(ξ,t)=1λ+(ξ)-λ-(ξ)(eλ+(ξ)t-eλ-(ξ)t),(15)H^(ξ,t)=1λ+(ξ)-λ-(ξ)(λ+(ξ)eλ-(ξ)t-λ-(ξ)eλ+(ξ)t).
We define G(x,t) and H(x,t) by
(16)G(x,t)=ℱ-1[G^(ξ,t)](x),(17)H(x,t)=ℱ-1[H^(ξ,t)](x),
respectively, where ℱ-1 denotes the inverse Fourier transform. Then, applying ℱ-1 to (13), we obtain
(18)u(t)=G(t)*u1+H(t)*u0.
By the Duhamel principle, we obtain the solution formula to (1), (2) as follows:
(19)u(t)=G(t)*u1+H(t)*u0+∫0t‍G(t-τ)*Δf(u)(τ)dτ.


## 3. Decay Property

The aim of this section is to establish decay estimates of the solution operatorsG(t) and H(t) appearing in (16) and (17), respectively.Lemma 1.
The solution of the problem (9), (10) satisfies
(20)(|ξ|2+|ξ|4)|u^(ξ,t)|2+|u^t(ξ,t)|2≤≤≤Ce-c|ξ|2t((|ξ|2+|ξ|4)|u^0(ξ)|2+|u^1(ξ)|2)
for ξ∈ℝn and t≥0.Proof.
Multiplying (9) by u^-t and taking the real part yield
(21)12ddt{|u^t|2+|ξ|2|u^|2+α|ξ|4|u^|2}+2b|ξ|2|u^t|2=0.
Multiplying (9) by u^- and taking the real part, we obtain
(22)12ddt{2b|ξ|2|u^|2+2Re(u^tu^-)}+++|ξ|2|u^|2+α|ξ|4|u^|2-|u^t|2=0.
Multiplying both sides of (22) by b|ξ|2 and summing up the resulting equation and (21) yield
(23)ddtE+F=0,
where
(24)E=12|u^t|2+12|ξ|2|u^|2+α2|ξ|4|u^|2+b2|ξ|4|u^|2+b|ξ|2Re(u^tu^-),F=b|ξ|2|u^t|2+b|ξ|4|u^|2+bα|ξ|6|u^|2.
A simple computation implies that
(25)CE0≤E≤CE0,
where
(26)E0=(|ξ|2+|ξ|4)|u^|2+|u^t|2.
Note that
(27)F≥c|ξ|2E0.
It follows from (25) that
(28)F≥c|ξ|2E.
Using (23) and (28), we get
(29)ddtE+c|ξ|2E≤0.
Thus
(30)E(ξ,t)≤e-c|ξ|2tE(ξ,0),
which together with (25) proves the desired estimates (20). Then we have completed the proof of the lemma.Lemma 2.
LetG^(ξ,t) and H^(ξ,t) be the fundamental solution of (8) in the Fourier space, which are given in (14) and (15), respectively. Then one has the estimates
(31)(|ξ|2+|ξ|4)|G^(ξ,t)|2+|G^t(ξ,t)|2≤Ce-c|ξ|2t,(32)(|ξ|2+|ξ|4)|H^(ξ,t)|2+|H^t(ξ,t)|2≤C(|ξ|2+|ξ|4)e-c|ξ|2t
for ξ∈ℝn and t≥0.Proof.
Firstly, we investigate the problem (8), (2) with u^0(ξ)=0; from (13), we obtain
(33)u^(ξ,t)=G^(ξ,t)u^1(ξ),u^t(ξ,t)=G^t(ξ,t)u^1(ξ).
Substituting the equalities into (20) with u^0(ξ)=0, we get (31).
In what follows, we consider the problem (8), (2) with u^1(ξ)=0; it follows from (13) that
(34)u^(ξ,t)=H^(ξ,t)u^0(ξ),u^t(ξ,t)=H^t(ξ,t)u^0(ξ).
Substituting the equalities into (20) with u^1(ξ)=0, we get the desired estimate (32). The lemma is proved.Lemma 3.
LetG(x,t) and H(x,t) be the fundamental solution of (8), which are given in (16) and (17), respectively. Let 1≤p≤2, and let k, j, and l be nonnegative integers. Then one has
(35)∥∂xkG(t)*ϕ∥L2≤≤C(1+t)-(n/2)(1/p-1/2)-(k-j)/2∥∂xjϕ∥W˙-1,p≤≤+Ce-ct∥∂xk+l-2ϕ∥L2,(36)∥∂xkH(t)*ϕ∥L2≤≤C(1+t)-(n/2)(1/p-1/2)-(k-j)/2∥∂xjϕ∥Lp≤≤+Ce-ct∥∂xk+lϕ∥L2
for 0≤j≤k, where k+l-2≥0 in (35). Similarly, one has
(37)∥∂xkGt(t)*ϕ∥L2≤≤C(1+t)-(n/2)(1/p-1/2)-(k+1-j)/2∥∂xjϕ∥W˙-1,p≤≤+Ce-ct∥∂xk+lϕ∥L2,(38)∥∂xkHt(t)*ϕ∥L2≤≤C(1+t)-(n/2)(1/p-1/2)-(k+1-j)/2∥∂xjϕ∥Lp≤≤+Ce-ct∥∂xk+l+2ϕ∥L2
for 0≤j≤k+1.Proof.
We only prove (35). By the Plancherel theorem, (31), and Hausdorff-Young inequality, we obtain
(39)∥∂xkG(t)*ϕ∥L22=∫|ξ|≤1‍|ξ|2|k||G^(ξ,t)|2|ϕ^(ξ)|2dξ++++++++++∫|ξ|≥1‍|ξ|2|k||G^(ξ,t)|2|ϕ^(ξ)|2dξ+++++++≤C∫|ξ|≤1‍|ξ|2|k|-2e-c|ξ|2t|ϕ^(ξ)|2dξ++++++++++Ce-ct∫|ξ|≥1‍|ξ|2k(|ξ|2+|ξ|4)-2|ϕ^(ξ)|2dξ+++++++≔𝒜1+𝒜2.
For the term 𝒜1, letting 1/p′+1/p=1, we have(40)𝒜1≤C∫|ξ|≤1‍|ξ|2k-2e-c|ξ|2t|ϕ^(ξ)|2dξ≤≤C∥|ξ|j-1ϕ^∥Lp′2(∫|ξ|≤1‍|ξ|2(k-j)pe-cq|ξ|2tdξ)1/p≤≤C(1+t)-n(1/p-1/2)-(k-j)∥∂xjϕ∥W˙-1,p2,
where we used the Hölder inequality with 2/p′+1/q=1 and the Hausdorff-Young inequality ∥v^∥Lp′≤C∥v∥Lp for v=(-Δ)-1/2∂xjϕ. On the other hand, we can estimate the term 𝒜2 simply as
(41)𝒜2≤Ce-ct∫|ξ|≥1‍|ξ|2k-4|ϕ^(ξ)|2dξ≤≤Ce-ct∫|ξ|≥1‍|ξ|2(k+l-2)|ϕ^(ξ)|2dξ≤≤Ce-ct∥∂xk+l-2ϕ∥L22,
where k+l-2≥0.
Combining the previous three inequalities yields (35). This completes the proof of Lemma 3.From Lemma3, we immediately have the following corollary.Corollary 4.
LetG(x,t) and H(x,t) be the fundamental solution of (8), which are given in (15) and (16), respectively. Let 1≤p≤2, and let k, j, and l be nonnegative integers. Then one has
(42)∥∂xkG(t)*Δg∥L2≤≤C(1+t)-(n/2)(1/p-1/2)-(k+1-j)/2∥∂xjg∥Lp≤≤+Ce-ct∥∂xk+lg∥L2
for 0≤k≤j+1. Also one has
(43)∥∂xkGt(t)*Δg∥L2≤≤C(1+t)-(n/2)(1/p-1/2)-(k+2-j)/2∥∂xjg∥Lp≤≤+Ce-ct∥∂xk+l+2g∥L2
for 0≤k≤j+2.


## 4. Global Existence and Asymptotic Behavior of Solutions to (1), (2)

The purpose of this section is to prove global existence and asymptotic behavior of solutions to the Cauchy problem (1), (2). We need the following lemma, which come from [18] (see also [19]).Lemma 5.
Assume thatf=f(v) is a smooth function. Suppose that f(v)=O(|v|1+θ) (θ≥1 is an integer) when |v|≤ν0. Then for integer m≥0, if v∈Wm,q(ℝn)⋂‍Lp(ℝn)⋂‍L∞(ℝn) and ∥v∥L∞≤ν0, then the following inequalities hold:
(44)∥∂xmf(v)∥Lr≤C∥v∥Lp∥∂xmv∥Lq∥v∥L∞θ-1,
where 1/r=1/p+1/q, 1≤p,q, r≤+∞.Theorem 6.
Letn≥1 and s≥max{0,[n/2]-1}. Suppose that f(u) is smooth and satisfies f(u)=O(u2) for u→0. Assume that u0∈Hs+2⋂‍L1, u1∈Hs⋂‍W˙-1,1. Put
(45)E0≔∥u0∥L1+∥u1∥W˙-1,1+∥u0∥Hs+2+∥u1∥Hs.
Then there exists a positive constant δ1 such that if E0≤δ1, the Cauchy problem (1), (2) has a unique global solution u(x,t) satisfying
(46)u∈C0([0,∞);Hs+2(ℝn))⋂‍C1([0,∞);Hs(ℝn)).
Moreover, the solution satisfies the decay estimate
(47)∥∂xku(t)∥L2≤CE0(1+t)-n/4-k/2,(48)∥∂xlut(t)∥L2≤CE0(1+t)-n/4-(k+1)/2,
where 0≤k≤s+2 in (47) and 0≤k≤s in (48).Proof.
The existence and uniqueness of small solutions can be proved by the contraction mapping principle. Here we only show the decay estimates (47) and (48) for the solution u of (19) satisfying ∥u(t)∥L∞≤M0 with some M0. To this end, we introduce the quantity
(49)M(t)=∑k=0s+1‍sup0≤τ≤t(1+τ)n/4+k/2∥∂xku(τ)∥L2.
Here we note that
(50)∥u(t)∥L∞≤CM(t)(1+t)-n/2,
provided that s≥max{0,[n/2]-1}. This follows from the Gagliardo-Nirenberg inequality ∥u∥L∞≤C∥∂xs0u∥L2θ∥u∥L21-θ with s0=[n/2]+1 and θ=n/2s0 and the definition of M(t) in (49).
Applying∂xk to (19) and taking the L2 norm, we obtain
(51)∥∂xku(t)∥L2≤∥∂xkG(t)*u1∥L2+∥∂xkH(t)*u0∥L2++++++++∫0t‍∥G(t-τ)*Δf(u(τ))∥L2dτ+++++≔I1+I2+J.
Firstly, we estimate I1. We apply (35) with p=1, j=0, and l=0(l=2 for k=0, l=1 for k=1). This yields
(52)I1≤C(1+t)-n/4-k/2∥u1∥W˙-1,1+Ce-ct∥∂x(k-2)+u1∥L2≤≤CE0(1+t)-n/4-k/2,
where (k-2)+=max{k-2,0}. Similarly, applying (36) with p=1, j=0, and l=0 to the term I2, we have
(53)I2≤C(1+t)-n/4-k/2∥u0∥L1+≤+Ce-ct∥∂xku0∥L2≤CE0(1+t)-n/4-k/2.
We estimate the nonlinear term J. We divide J into two parts and write J=J1+J2, where J1 and J2 are corresponding to the time intervals [0,t/2] and [t/2,t], respectively. For the term J1, we apply (42) with p=1, j=0, and l=0. This yields
(54)J1≤C∫0t/2‍(1+t-τ)-n/4-(k+1)/2∥f(u)(τ)∥L1dτ++++C∫0t/2‍e-c(t-τ)∥∂xkf(u)(τ)∥L2dτ+≔J11+J12.
Here we see that ∥f(u)∥L1≤C∥u∥L22 by Lemma 5. Thus we have ∥f(u)(τ)∥L1≤CM(t)2(1+τ)-n/2. Therefore we can estimate the term J11 as
(55)J11≤CM(t)2∫0t/2(1+t-τ)-n/4-(k+1)/2(1+τ)-n/2dτ≤≤CM(t)2(1+t)-n/4-(k+1)/2∫0t/2(1+τ)-n/2dτ≤≤CM(t)2(1+t)-n/4-k/2η(t),
where
(56)η(t)={1,n=1,(1+t)-1/2log(2+t),n=2,(1+t)-1/2,n≥3.
On the other hand, we have ∥∂xkf(u)∥L2≤C∥u∥L∞∥∂xku∥L2 by Lemma 5. Therefore, using (50), we find that ∥∂xkf(u)(τ)∥L2≤CM(t)2(1+τ)-n/4-k/2-n/2. Consequently, we can estimate the term J12 as
(57)J12≤CM(t)2∫0t/2‍e-c(t-τ)(1+τ)-n/4-k/2-n/2dτ≤CM(t)2e-ct.
Finally, we estimate the term J2 on the time interval [t/2,t]. Applying (42) with p=2, j=k, and l=0 and using ∥∂xkf(u)(τ)∥L2≤CM(t)2(1+τ)-n/4-k/2-n/2, we obtain
(58)J2≤C∫t/2t‍(1+t-τ)-1/2∥∂xkf(u)(τ)∥L2dτ+≤+C∫t/2t‍e-c(t-τ)∥∂xkf(u)(τ)∥L2dτ≤≤CM(t)2∫t/2t(1+t-τ)-1/2(1+τ)-n/4-k/2-n/2dτ≤≤CM(t)2(1+t)-n/4-k/2-(n-1)/2.
Thus we have shown that
(59)J≤CM(t)2(1+t)-n/4-k/2η(t).
Substituting all these estimates into (51), we obtain
(60)(1+t)n/4+k/2∥∂xku(t)∥L2≤CE0+CM(t)2
for 0≤k≤s+2. Consequently, we have M(t)≤CE0+CM(t)2, from which we can deduce M(t)≤CE0, provided that E0 is suitably small. This proves the decay estimate (47).
In what follows, we prove the decay estimate (48) for the time derivative ut. For this purpose, we differentiate (19) with respect to t to obtain
(61)ut(t)=Gt(t)*u1+Ht(t)*u1+∫0t‍Gt(t-τ)*Δf(u)(τ)dτ.
Applying ∂xk to (61) and taking the L2 norm, we have
(62)∥∂xkut(t)∥L2≤∥∂xkGt(t)*u1∥L2+∥∂xkHt(t)*u0∥L2++++++++∫0t‍∥∂xkGt(t-τ)*Δf(u)(τ)∥L2dτ+++++≔I1′+I2′+J′,
where 0≤k≤s. For the term I1′, we apply (37) with p=1, j=0, and l=0 to get
(63)I1′≤C(1+t)-n/4-(k+1)/2∥u1∥W˙-1,1+≤+e-ct∥∂xku1∥L2≤CE0(1+t)-n/4-(k+1)/2.
Also, for the term I2′, applying (38) with p=1, j=0, and l=0, we have
(64)I2′≤C(1+t)-n/4-(k+1)/2∥u0∥L1+e-ct∥∂xk+2u0∥L2≤CE0(1+t)-n/4-(k+1)/2.
To estimate the nonlinear term J′, we divide J1′ as J′=J1′+J2′, where J1′ and J2′ correspond to the time intervals [0,t/2] and [t/2,t], respectively. For the term J1′, we apply (43) with p=1, j=0, and l=0. This yields
(65)J1′≤C∫0t/2‍(1+t-τ)-n/4-(k+2)/2∥f(u)(τ)∥L1dτ++++C∫0t/2‍e-c(t-τ)∥∂xk+2f(u)(τ)∥L2dτ+≔J11′+J12′.
Since ∥f(u)(τ)∥L1≤CM(t)2(1+τ)-n/2 as before, we can estimate the term J11′ as
(66)J11′≤CM(t)2∫0t/2‍(1+t-τ)-n/4-(k+2)/2(1+τ)-n/2dτ≤≤CM(t)2(1+t)-n/4-(k+1)/2η(t).
Also, the term J12′ is estimated similarly as before, and we have J12′≤CM(t)2e-ct. Finally, we estimate the term J2′ by applying (43) with p=2, j=k+2, and l=0 and obtain
(67)J2′≤C∫t/2t‍∥∂xk+2f(u)(τ)∥L2dτ+≤+C∫t/2t‍e-c(t-τ)∥∂xk+2f(u)(τ)∥L2dτ≤≤CM(t)2∫t/2t‍(1+τ)-n/4-(k+2)/2-n/2dτ≤≤CM(t)2(1+t)-n/4-(k+1)/2-(n-1)/2,
where we used the estimate ∥∂xk+2f(u)(τ)∥L2≤CM(t)2(1+τ)-n/4-(k+2)/2-n/2. Consequently we have shown that
(68)J′≤CM(t)2(1+t)-n/4-(k+1)/2η(t).
Substituting all these estimates together with the previous estimate M(t)≤CE0 into (62), we arrive at the desired estimate (48) for 0≤k≤s. This completes the proof of Theorem 6.The previous proof of Theorem6 shows that when n≥2, the solution u to the integral equation (19) is asymptotic to the linear solution uL given by the formula uL(t)≔G(t)*u1+H(t)*u0 in (18) as t→∞. This result is stated as follows.Corollary 7.
Letn≥2 and assume the same conditions of Theorem 6. Then the solution u of the problem (1), (2), which is constructed in Theorem 6, can be approximated by the solution uL to the linearized problem (8), (2) as t→∞. More precisely, one has the following asymptotic relations:
(69)∥∂xk(u-uL)(t)∥L2≤CE02(1+t)-n/4-k/2η(t),∥∂xk(u-uL)t(t)∥L2≤CE02(1+t)-n/4-(k+1)/2η(t)
for 0≤k≤s+2 and 0≤k≤s, respectively, where uL(t)≔G(t)*u1+H(t)*u0 is the linear solution and η(t) is defined in (56).


## 5. Asymptotic Linear Profile

In the previous section, we have shown that the solutionu to the problem (1), (2) can be approximated by the linear solution uL. The aim of this section is to derive a simpler asymptotic profile of the linear solution uL.In Section2, we derive the solution formula uL(t)=G(t)*u1+H(t)*u0 to the problem (8), (9). In the Fourier space, we have u^L(ξ,t)=G^(ξ,t)u^1(ξ)+H^(ξ,t)u^0(ξ), where G^(ξ,t) and H^(ξ,t) are given explicitly in (14) and (15), respectively. First we give the asymptotic expansions of G^(ξ,t) and H^(ξ,t) for ξ→0. Applying the Taylor expansion to (12), we see that
(70)λ±(ξ)=-b|ξ|2±i|ξ|1+(α-b2)|ξ|2=-b|ξ|2±i|ξ|(1+12(α-b2)|ξ|2+O(|ξ|4))=±i|ξ|-b|ξ|2+O(|ξ|3),1λ+-λ-=12i|ξ|1+(α-b2)|ξ|2=12i|ξ|{1-12(α-b2)|ξ|2+O(ξ4)}.
Inserting the previous equalities into (14) and (15), respectively, we arrive at
(71)G^(ξ,t)=eλ+t-eλ-tλ+-λ-=12i|ξ|{e-b|ξ|2t(e|ξ|ti-e-|ξ|ti)++++e-b|ξ|2t(O(|ξ|)+O(|ξ|3t))},H^(ξ,t)=λ+eλ-t-λ-eλ+tλ+-λ-=12e-b|ξ|2t(e|ξ|ti+e-|ξ|ti)+e-b|ξ|2t{O(|ξ|)+O(|ξ|3t)}
for ξ→0. Taking
(72)G^0(ξ,t)=12i|ξ|e-b|ξ|2t(ei|ξ|t-e-i|ξ|t),H^0(ξ,t)=12e-b|ξ|2t(ei|ξ|t+e-i|ξ|t).
Then we find that
(73)|(G^-G^0)(ξ,t)|≤Ce-c|ξ|2t,|(H^-H^0)(ξ,t)|≤C|ξ|e-c|ξ|2t
for |ξ|≤r0, where r0 is a small positive constant.We now defineu-L by
(74)u-L(t)=G0(t)*u1+H0(t)*u0.
Then u-L gives a asymptotic profile of the linear solution uL. In fact we have the following.Theorem 8 (n≥1 and s≥0).
Assume thatu0∈Hs+2∩L1 and u1∈Hs∩W˙-1,1, and put E0=∥u0∥L1+∥u1∥W˙-1,1+∥u0∥Hs+2+∥u1∥Hs. Let uL be the linear solution, and let u-L be defined by (74). Then one has
(75)∥∂xk(uL-u-L)(t)∥L2≤CE0(1+t)-n/4-(k+1)/2
for 0≤k≤s+2.Proof.
Note that(uL-u-L)(t)=(G-G0)(t)*u1+(H-H0)(t)*u0, so for the proof of (75), it suffices to show the following estimates:
(76)∥∂xk(G-G0)(t)*u1∥L2≤≤≤C(1+t)-(n/2)(1/p-1/2)-(k+1-j)/2∥∂xju1∥W˙-1,p++++Ce-ct∥∂xk+l-2u1∥L2,∥∂xk(H-H0)(t)*u0∥L2≤≤≤C(1+t)-(n/2)(1/p-1/2)-(k+1-j)/2∥∂xju0∥Lp++++Ce-ct∥∂xk+lu0∥L2,
where 1≤p≤2 and k, j, and l are nonnegative integers such that 0≤j≤k+1; we assumed that k+l-2≥0 in the first estimate. These estimates can be proved similarly as in the proof of Lemma 3 by using (73) for |ξ|≤r0 and (31), (32), and (72) for |ξ|≥r0. Here we omit the details.



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*Source: 364165-2013-08-01.xml*