# Asymptotic Behavior of Global Solutions to the Boussinesq Equation in Multidimensions

Authors: Yu-Zhu Wang; Qingnian Zhang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The Cauchy problem for the Boussinesq equation in multidimensions is investigated. We prove
the asymptotic behavior of the global solutions provided that the initial data are suitably small.
Moreover, our global solutions can be approximated by the solutions to the corresponding linear
equation as time tends to infinity when the dimension of spacen≥3.


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## Body

## 1. Introduction

We investigate the Cauchy problem of the following damped Boussinesq equation in multidimensions:(1)utt-aΔutt-2bΔut-αΔ3u+βΔ2u-Δu=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,  a,b,α,  and β are positive constants. The nonlinear term f(u)=O(u2).Whenf(u)=u2, (1) has been studied by several authors. The authors investigated the first initial boundary value problem for (1) in a unit circle (see [1]). The existence and the uniqueness of strong solution were established and the solutions were constructed in the form of series in the small parameter present in the initial conditions. The long-time asymptotic was also obtained in the explicit form. The authors considered the initial-boundary value problem for (1) in the unit ball B⊂ℝ3, similar results were established in [2].Recently, Wang [3] proved the global existence and asymptotic decay of solutions to the problem (1), (2). Their proof is based on the contraction mapping principle and makes use of the sharp decay estimates for the linearized problem. The main purpose of this paper is to establish the following optimal decay estimate of solutions to (1) and (2) by constructing the antiderivatives conditions:
(3)u1(x)=∂x1v1(x).
Then we obtain a better decay rate of solutions than the previous one in [3]. Moreover, our global solutions can be approximated by the solutions to the corresponding linear equation. The decay estimate is said to be optimal because we have used the sharp decay estimates for the solution operators G(x,t) and H(x,t), which are defined by (15) and (16), respectively. Since the solution operator G(x,t) has singularity, therefore, we construct the antiderivatives conditions u1(x)=∂x1v1(x) and eliminate the singularity and obtain the same decay estimate for the solution operators H(x,t). For details; see Lemma 4. The study of the global existence and asymptotic behavior of solutions to wave equations has a long history. We refer to [4–10] for wave equations. Now we state our results as follows.Theorem 1.
Lets≥[n/2]-1 and let n≥2. Assume that u0∈Hs+2(ℝn),v1∈Hs+1(ℝn). Put
(4)E0=∥u0∥Hs+2+∥v1∥Hs+1.
If E0 is suitably small, the Cauchy problem (1), (2) has a unique global solution u(x,t) satisfying
(5)u∈C([0,∞);Hs+2)⋂‍C1([0,∞);Hs).
Moreover, the solution satisfies the decay estimate:
(6)∥∂xku(t)∥L2≤CE0(1+t)-(k/2),(7)∥∂xkut(t)∥L2≤CE0(1+t)-((k+1)/2)
for 0≤k≤s+2 in (6) and 0≤k≤s in (7).From the proof of Theorem1, we have the following corollary immediately.Corollary 2.
Letn≥3 and assume the same conditions of Theorem 1. Then the solution u of the problem (1), (2), which is constructed in Theorem 1, can be approximated by the linear solution uL as t→∞. In fact, we have
(8)∥∂xk(u-uL)(t)∥L2≤CE02(1+t)-(k/2)η(t),∥∂xk(u-uL)t(t)∥L2≤CE02(1+t)-((k+1)/2)η(t)
for 0≤k≤s+1 and 0≤k≤s, respectively, where uL(t)∶=G(t)*∂x1v1+H(t)*u0 is the linear solution and η(t)=(1+t)-((n-2)/4). Here G(t) and H(t) are given by (15) and (16), respectively.Notations. For 1≤p≤∞, Lp=Lp(ℝn) denotes the usual Lebesgue space with the norm ∥·∥Lp. The usual Sobolev space of order s is defined by Ws,p=(I-Δ)-(s/2)Lp with the norm ∥f∥Ws,p=∥(I-Δ)s/2f∥Lp. The corresponding homogeneous Sobolev space of order s is defined by W˙s,p=(-Δ)-(s/2)Lp with the norm ∥f∥W˙s,p=∥(-Δ)s/2f∥Lp; when p=2, we write Hs=Ws,2 and H˙s=W˙s,2. We note that Ws,p=Lp∩W˙s,p for s≥0.The plan of the paper is arranged as follows. In Section2 we derive the solution formula of the problem (1), (2) and prove the decay property of the solution operators appearing in the solution formula. Then, in Sections 3, we prove the optimal asymptotic decay of solutions to the problem (1), (2).


## 2.  Decay Property

The aim of this section is to derive the solution formula for the problem (1), (2). We first investigate the linear equation of (1):
(9)utt-aΔutt-2bΔut-αΔ3u+βΔ2u-Δu=0.
With the initial data (2). Taking the Fourier transform, we have
(10)(1+a|ξ|2)u^tt+2b|ξ|2u^t+(α|ξ|6+β|ξ|4+|ξ|2)u^=0;(11)t=0:u^=u^0(ξ),u^t=iξ1v^1(ξ).
The characteristic equation of (10) is
(12)(1+a|ξ|2)λ2+2b|ξ|2λ+α|ξ|6+β|ξ|4+|ξ|2=0.
Let λ=λ±(ξ) be the corresponding eigenvalues of (12), we obtain
(13)λ±(ξ)=-b|ξ|2±|ξ|-1-(a+β-b2)|ξ|2-(α+aβ)|ξ|4-aα|ξ|61+a|ξ|2.
The solution to the problem (10), (11) is given in the form
(14)u^(ξ,t)=G^(ξ,t)iξ1v^1(ξ)+H^(ξ,t)u^0(ξ),
where
(15)G^(ξ,t)=1λ+(ξ)-λ-(ξ)(eλ+(ξ)t-eλ-(ξ)t),(16)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 G(x,t)=F-1[G^(ξ,t)](x) and H(x,t)=F-1[H^(ξ,t)](x), respectively, where F-1 denotes the inverse Fourier transform. Then, applying F-1 to (14), we obtain
(17)u(t)=G(t)*∂x1v1+H(t)*u0.
By the Duhamel principle, we obtain the solution formula to (1), (2) as
(18)u(t)=G(t)*∂x1v1+H(t)*u0+∫0t‍G(t-τ)*(I-aΔ)-1Δf(u)(τ)dτ.
In what follows, the aim is to establish decay estimates of the solution operators G(t) and H(t) appearing in the solution formula (18). Firstly, we state the pointwise estimate of solutions in the Fourier space. The result can be found in [3].Lemma 3.
The solution of the problem (10), (11) satisfies
(19)|ξ|2(1+|ξ|2)|u^(ξ,t)|2+|u^t(ξ,t)|2≤Ce-cω(ξ)t(|ξ|2(1+|ξ|2)|u^0(ξ)|2+|ξ1|2|v^1(ξ)|2),
for ξ∈ℝn and t≥0, where ω(ξ)=|ξ|2/(1+|ξ|2).From Lemma3, we immediately get the following.Lemma 4.
LetG^(ξ,t) and H^(ξ,t) be the fundamental solution of (10) in the Fourier space, which are given in (15) and (16), respectively. Then we have the estimates
(20)|ξ|2(1+|ξ|2)|G^(ξ,t)|2+|G^t(ξ,t)|2≤Ce-cω(ξ)t,(21)|ξ|2(1+|ξ|2)|H^(ξ,t)|2+|H^t(ξ,t)|2≤C|ξ|2(1+|ξ|2)e-cω(ξ)t
for ξ∈ℝn and t≥0, where ω(ξ)=|ξ|2/(1+|ξ|2).Lemma 5.
Letk,j,l be nonnegative integers and let 1≤p≤2. Then we have
(22)∥∂xkG(t)*∂x1ϕ∥L2≤C(1+t)-(n/2)((1/p)-(1/2))-((k-j)/2)∥∂xjϕ∥Lp+Ce-ct∥∂xk+l-1ϕ∥L2,(23)∥∂xkH(t)*ψ∥L2≤C(1+t)-(n/2)((1/p)-(1/2))-((k-j)/2)∥∂xjψ∥Lp+Ce-ct∥∂xk+lψ∥L2
for 0≤j≤k, where k+l-1≥0 in (22). Similarly, we have
(24)∥∂xkGt(t)*∂x1ϕ∥L2≤C(1+t)-(n/2)((1/p)-(1/2))-((k+1-j)/2)∥∂xjϕ∥Lp+Ce-ct∥∂xk+l+1ϕ∥L2,(25)∥∂xkHt(t)*ψ∥L2≤C(1+t)-(n/2)((1/p)-(1/2))-((k+1-j)/2)∥∂xjψ∥Lp+Ce-ct∥∂xk+l+2ψ∥L2
for 0≤j≤k+1.Proof.
We only prove (22). By the Plancherel theorem and (20), we obtain
(26)∥∂xkG(t)*∂x1ϕ∥L22=∫|ξ|≤1‍|ξ|2k|G^(ξ,t)|2|ξ1|2|ϕ^(ξ)|2dξ+∫|ξ|≥1‍|ξ|2k|G^(ξ,t)|2|ξ1|2|ϕ^(ξ)|2dξ≤C∫|ξ|≤1‍|ξ|2ke-c|ξ|2t|ϕ^(ξ)|2dξ+Ce-ct∫|ξ|≥1‍|ξ|2k+2(|ξ|2(1+|ξ|2))-1|ϕ^(ξ)|2dξ=:I1+I2.
For the term I1, letting 1/p′+1/p=1, we have
(27)I1≤C∫|ξ|≤1‍|ξ|2ke-c|ξ|2t|ϕ^(ξ)|2dξ≤C∥|ξ|jϕ^∥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∥Lp2,
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 I2 simply as
(28)I2≤Ce-ct∫|ξ|≥1‍|ξ|2k-2|ϕ^(ξ)|2dξ≤Ce-ct∫|ξ|≥1‍|ξ|2(k+l-1)|ϕ^(ξ)|2dξ≤Ce-ct∥∂xk+l-1ϕ∥L22,
where k+l-1≥0. Combining (26)–(28) yields (22). We have completed the proof of the Lemma.Similar to the proof of Lemma5, it is not difficult to get the following.Lemma 6.
Let1≤p≤2 and let k,j,l be nonnegative integers. Then we have the following estimate:
(29)∥∂xkG(t)*(I-aΔ)-1Δg∥L2≤C(1+t)-(n/2)((1/p)-(1/2))-((k+1-j)/2)∥∂xjg∥Lp+Ce-ct∥∂xk+lg∥L2
for 0≤j≤k+1. Similarly, we have
(30)∥∂xkGt(t)*(I-aΔ)-1Δg∥L2≤C(1+t)-(n/2)((1/p)-(1/2))-((k+2-j)/2)∥∂xjg∥Lp+Ce-ct∥∂xk+lg∥L2
for 0≤j≤k+2.


## 3.  Proof of Main Result

In order to prove optimal decay estimate of solutions to the Cauchy problem (1), (2). We need the following Lemma, which comes from [11] (see also [12]).Lemma 7.
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:
(31)∥∂xmf(v)∥Lr≤C∥v∥Lp∥∂xmv∥Lq∥v∥L∞θ-1,
where 1/r=(1/p)+(1/q),1≤p,q,r≤+∞.Proof of Theorem1.
We can prove the existence and uniqueness of small solutions by the contraction mapping principle. Here we only show the decay estimates (6) and (7) for the solution u of (18) satisfying ∥u(t)∥L∞≤M0 with some M0. Firstly, we introduce the quantity:
(32)𝒲(t)=∑k=0s+2‍sup0≤τ≤t(1+τ)k/2∥∂xku(τ)∥L2.
We apply the Gagliardo-Nirenberg inequality. This yields
(33)∥u∥L∞≤C∥∂xs0u∥L2θ∥u∥L21-θ,
where s0=[n/2]+1 and θ=n/2s0. It follows from the definition of 𝒲(t) in (32) that
(34)∥u(t)∥L∞≤C𝒲(t)(1+t)-(n/4),
provided that s≥[n/2]-1. Differentiating (18) k times with respect to x and taking the L2 norm, we obtain
(35)∥∂xku(t)∥L2≤∥∂xkG(t)*∂x1v1∥L2+∥∂xkH(t)*u0∥L2+∫0t∥∂xkG(t-τ)*(I-aΔ)-1Δf(u(τ))∥L2‍dτ=I1+I2+J.
Firstly, we estimate I1. We get from (22), with p=2,j=0, and l=0(l=1 for k=0),
(36)I1≤C(1+t)-(k/2)∥v1∥L2+Ce-ct∥∂x(k-1)+v1∥L2≤CE0(1+t)-(k/2),
where (k-1)+=max{k-1,0}. By using (23) with p=2, j=0, and l=0 to the term I2, we obtain
(37)I2≤C(1+t)-(k/2)∥u0∥L2+Ce-ct∥∂xku0∥L2≤CE0(1+t)-(k/2).
Next, we estimate 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 J1, making use of (29) with p=2, j=0, and l=0, we arrive at
(38)J1≤C∫0t/2‍(1+t-τ)-((k+1)/2)∥f(u)(τ)∥L2dτ+C∫0t/2‍e-c(t-τ)∥∂xkf(u)(τ)∥L2dτ.
By Lemma 7, we have the estimates ∥f(u)∥L2≤C∥u∥L∞∥u∥L2 and ∥∂xkf(u)∥L2≤C∥u∥L∞∥∂xku∥L2. Thus by (34), we have
(39)∥f(u)(τ)∥L2≤C𝒲(t)2(1+τ)-(n/4),(40)∥∂xkf(u)(τ)∥L2≤C𝒲(t)2(1+τ)-((n/4)-(k/2)).
Inserting (39) and (40) into (38) yields
(41)J1≤C𝒲(t)2∫0t/2‍(1+t-τ)-((k+1)/2)(1+τ)-(n/4)dτ+C𝒲(t)2∫0t/2‍e-c(t-τ)(1+τ)-((n/4)-(k/2))dτ≤C𝒲(t)2(1+t)-(k/2)η(t),
where η(t)=(1+t)-((n-2)/4). Here we assumed n≥2. For J2, exploiting (29) with p=2, j=k, and l=0 and using (40), we deduce that
(42)J2≤C∫t/2t‍(1+t-τ)-(1/2)∥∂xkf(u)(τ)∥L2dτ+C∫t/2t‍e-c(t-τ)∥∂xkf(u)(τ)∥L2dτ≤C𝒲(t)2∫t/2t‍(1+t-τ)-(1/2)(1+τ)-((n/4)-(k/2))dτ≤C𝒲(t)2(1+t)-(k/2)η(t).
Equations (41) and (42) give
(43)J≤C𝒲(t)2(1+t)-((n/4)-(k/2))η(t).
Inserting (36), (37), and (43) into (35), we obtain
(44)(1+t)k/2∥∂xku(t)∥L2≤CE0+C𝒲(t)2
for 0≤k≤s+2. Consequently, we have 𝒲(t)≤CE0+C𝒲(t)2, from which we can deduce 𝒲(t)≤CE0, provided that E0 is suitably small. This proves the decay estimate (6).
In what follows, we prove (7). Differentiating (18) with respect to t and then differentiating the resulting equation k times with respect to x, we have
(45)∂xkut(t)=∂xkGt(t)*∂x1v1+∂xkHt(t)*u0+∫0t‍∂xkGt(t-τ)*(I-aΔ)-1Δf(u)(τ)dτ.
From (45) and Minkowski inequality, we obtain
(46)∥∂xkut(t)∥L2≤∥∂xkGt(t)*∂x1v1∥L2+∥∂xkHt(t)*u0∥L2+∫0t∥∂xkGt(t-τ)*(I-aΔ)-1Δf(u(τ))∥L2‍dτ=K1+K2+L.
It follows from (24) that
(47)K1≤C(1+t)-((k+1)/2)∥v1∥L2+Ce-ct∥∂xk+1v1∥L2≤CE0(1+t)-((k+1)/2).
By using (25), we get
(48)K2≤C(1+t)-((k+1)/2)∥u0∥L2+Ce-ct∥∂xk+2u0∥L2≤CE0(1+t)-((k+1)/2).
Finally, we estimate L. Dividing L into two parts and writing L=L1+L2, where L1 and L2 are corresponding to the time intervals [0,t/2] and [t/2,t], respectively. Firstly, we estimate the term L1, applying (30) with p=2, j=0, and l=0 and (39), (40), we arrive at
(49)L1≤C∫0t/2‍(1+t-τ)-((k+2)/2)∥f(u)(τ)∥L2dτ+C∫0t/2‍e-c(t-τ)∥∂xkf(u)(τ)∥L2dτ≤C𝒲2(t)∫0t/2‍(1+t-τ)-((k+2)/2)(1+τ)-(n/4)dτ+C𝒲2(t)∫0t/2‍e-c(t-τ)(1+τ)-((n/4)-(k/2))dτ≤C𝒲2(t)(1+t)-((k+1)/2)η(t).
Next, for the term L1, it follows from (30) with p=2,  k=0, and l=0 and (40) that
(50)L2≤C∫t/2t‍(1+t-τ)-1∥∂xkf(u)(τ)∥L2dτ+C∫t/2t‍e-c(t-τ)∥∂xkf(u)(τ)∥L2dτ≤C𝒲2(t)∫t/2t‍(1+t-τ)-1(1+τ)-((n/4)-(k/2))dτ+C𝒲2(t)∫t/2t‍e-c(t-τ)(1+τ)-((n/4)-(k/2))dτ≤C𝒲2(t)(1+t)-((k+1)/2)η(t).
Collecting (46)–(50), which yields
(51)∥∂xkut(t)∥L2≤CE0(1+t)-((k+1)/2)+C𝒲2(t)(1+t)-((k+1)/2)η(t).
Substituting the estimate 𝒲(t)(t)≤CE0 into (51), we arrive at the desired estimate (7) for 0≤k≤s. This completes the proof of Theorem 1.



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