Asymptotic Behavior of Global Solutions to the Boussinesq Equation in Multidimensions

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 space 𝑛≥3.

1. Introduction

We investigate the Cauchy problem of the following damped Boussinesq equation in multidimensions: 𝑢_𝑡𝑡−𝑎Δ𝑢_𝑡𝑡−2𝑏Δ𝑢_𝑡−𝛼Δ³𝑢+𝛽Δ²𝑢−Δ𝑢=Δ𝑓(𝑢) (1) with the initial value 𝑡=0∶𝑢=𝑢₀(𝑥),𝑢_𝑡=𝑢₁(𝑥). (2) Here 𝑢=𝑢(𝑥,𝑡) is the unknown function of 𝑥=(𝑥₁,…,𝑥_𝑛)∈ℝ^𝑛 and 𝑡>0,  𝑎,𝑏,𝛼,  and 𝛽 are positive constants. The nonlinear term 𝑓(𝑢)=𝑂(𝑢²).

When 𝑓(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: 𝑢₁(𝑥)=𝜕_𝑥1𝑣₁(𝑥). (3) 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. Let 𝑠≥[𝑛/2]−1 and let 𝑛≥2. Assume that 𝑢₀∈𝐻^𝑠+2(ℝ^𝑛),𝑣₁∈𝐻^𝑠+1(ℝ^𝑛). Put 𝐸₀=𝑢_0𝐻𝑠+2+𝑣_1𝐻𝑠+1. (4) If 𝐸₀ is suitably small, the Cauchy problem (1), (2) has a unique global solution 𝑢(𝑥,𝑡) satisfying 𝑢∈𝐶[0,∞);𝐻^𝑠+2𝐶¹[0,∞);𝐻^𝑠. (5) Moreover, the solution satisfies the decay estimate: 𝜕^𝑘_𝑥𝑢(𝑡)_𝐿2≤𝐶𝐸₀(1+𝑡)^−(𝑘/2),𝜕^𝑘_𝑥𝑢_𝑡(𝑡)_𝐿2≤𝐶𝐸₀(1+𝑡)^{−((𝑘+1)/2)} (6)(7) for 0≤𝑘≤𝑠+2 in (6) and 0≤𝑘≤𝑠 in (7).

From the proof of Theorem 1, we have the following corollary immediately.

Corollary 2. Let 𝑛≥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 𝑢_𝐿 as 𝑡→∞. In fact, we have 𝜕^𝑘_𝑥𝑢−𝑢_𝐿(𝑡)_𝐿2≤𝐶𝐸²₀(1+𝑡)^−(𝑘/2)𝜂(𝑡),𝜕^𝑘_𝑥𝑢−𝑢_𝐿𝑡(𝑡)_𝐿2≤𝐶𝐸²₀(1+𝑡)^{−((𝑘+1)/2)}𝜂(𝑡) (8) for 0≤𝑘≤𝑠+1 and 0≤𝑘≤𝑠, respectively, where 𝑢_𝐿(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≤∞, 𝐿^𝑝=𝐿^𝑝(ℝ^𝑛) denotes the usual Lebesgue space with the norm ‖⋅‖_𝐿^𝑝. The usual Sobolev space of order 𝑠 is defined by 𝑊^𝑠,𝑝=(𝐼−Δ)^−(𝑠/2)𝐿^𝑝 with the norm ‖𝑓‖_{𝑊𝑠,𝑝}=‖(𝐼−Δ)^𝑠/2𝑓‖_𝐿𝑝. The corresponding homogeneous Sobolev space of order 𝑠 is defined by 𝑊̇^𝑠,𝑝=(−Δ)^−(𝑠/2)𝐿^𝑝 with the norm ‖𝑓‖_{̇𝑊𝑠,𝑝}=‖(−Δ)^𝑠/2𝑓‖_𝐿𝑝; when 𝑝=2, we write 𝐻^𝑠=𝑊^𝑠,2 and 𝐻̇^𝑠=𝑊̇^𝑠,2. We note that 𝑊^𝑠,𝑝=𝐿^𝑝∩𝑊̇^𝑠,𝑝 for 𝑠≥0.

The plan of the paper is arranged as follows. In Section 2 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): 𝑢_𝑡𝑡−𝑎Δ𝑢_𝑡𝑡−2𝑏Δ𝑢_𝑡−𝛼Δ³𝑢+𝛽Δ²𝑢−Δ𝑢=0. (9) With the initial data (2). Taking the Fourier transform, we have 1+𝑎𝜉²𝑢_𝑡𝑡+2𝑏𝜉²𝑢_𝑡+𝛼𝜉⁶+𝛽𝜉⁴+𝜉²𝑢𝑢=0;𝑡=0∶𝑢=₀(𝜉),𝑢_𝑡=𝑖𝜉₁𝑣₁̂(𝜉). (10)(11) The characteristic equation of (10) is 1+𝑎𝜉²𝜆²+2𝑏𝜉²𝜆+𝛼𝜉⁶+𝛽𝜉⁴+𝜉²=0. (12) Let 𝜆=𝜆_±(𝜉) be the corresponding eigenvalues of (12), we obtain 𝜆_±(𝜉)=−𝑏𝜉²±𝜉−1−𝑎+𝛽−𝑏²𝜉²−𝛼+𝑎𝛽𝜉⁴−𝑎𝛼𝜉⁶1+𝑎𝜉². (13) The solution to the problem (10), (11) is given in the form 𝑢(𝜉,𝑡)=𝐺(𝜉,𝑡)𝑖𝜉₁𝑣₁̂(𝜉)+𝐻(𝜉,𝑡)𝑢₀(𝜉), (14) where 𝐺()=1𝜉,𝑡𝜆₊(𝜉)−𝜆₋(𝜉)𝑒^{𝜆+(𝜉)𝑡}−𝑒^{𝜆−(𝜉)𝑡},𝐻()=1𝜉,𝑡𝜆₊(𝜉)−𝜆₋(𝜉)𝜆₊(𝜉)𝑒^{𝜆−(𝜉)𝑡}−𝜆₋(𝜉)𝑒^{𝜆+(𝜉)𝑡}. (15)(16) 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 𝑢(𝑡)=𝐺(𝑡)∗𝜕_𝑥1𝑣₁+𝐻(𝑡)∗𝑢₀. (17) By the Duhamel principle, we obtain the solution formula to (1), (2) as 𝑢(𝑡)=𝐺(𝑡)∗𝜕_𝑥1𝑣₁+𝐻(𝑡)∗𝑢₀+^𝑡₀𝐺(𝑡−𝜏)∗(𝐼−𝑎Δ)⁻¹Δ𝑓(𝑢)(𝜏)𝑑𝜏. (18) 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 𝜉²1+𝜉²𝑢(𝜉,𝑡)²+𝑢_𝑡(𝜉,𝑡)²≤𝐶𝑒^{−𝑐𝜔(𝜉)𝑡}𝜉²1+𝜉²𝑢₀(𝜉)²+𝜉₁²𝑣₁̂(𝜉)², (19) for 𝜉∈ℝ^𝑛 and 𝑡≥0, where 𝜔(𝜉)=|𝜉|²/(1+|𝜉|²).

From Lemma 3, we immediately get the following.

Lemma 4. Let 𝐺(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 𝜉²1+𝜉²𝐺(𝜉,𝑡)²+𝐺_𝑡(𝜉,𝑡)²≤𝐶𝑒^{−𝑐𝜔(𝜉)𝑡},𝜉²1+𝜉²𝐻(𝜉,𝑡)²+𝐻_𝑡(𝜉,𝑡)²≤𝐶𝜉²1+𝜉²𝑒^{−𝑐𝜔(𝜉)𝑡} (20)(21) for 𝜉∈ℝ^𝑛 and 𝑡≥0, where 𝜔(𝜉)=|𝜉|²/(1+|𝜉|²).

Lemma 5. Let 𝑘,𝑗,𝑙 be nonnegative integers and let 1≤𝑝≤2. Then we have 𝜕^𝑘_𝑥𝐺(𝑡)∗𝜕_𝑥1𝜙_𝐿2≤𝐶(1+𝑡)^{−(𝑛/2)((1/𝑝)−(1/2))−((𝑘−𝑗)/2)}𝜕^𝑗_𝑥𝜙_𝐿𝑝+𝐶𝑒^−𝑐𝑡𝜕_𝑥^{𝑘+𝑙−1}𝜙_𝐿2,𝜕^𝑘_𝑥𝐻(𝑡)∗𝜓_𝐿2≤𝐶(1+𝑡)^{−(𝑛/2)((1/𝑝)−(1/2))−((𝑘−𝑗)/2)}𝜕^𝑗_𝑥𝜓_𝐿𝑝+𝐶𝑒^−𝑐𝑡𝜕_𝑥^𝑘+𝑙𝜓_𝐿2 (22)(23) for 0≤𝑗≤𝑘, where 𝑘+𝑙−1≥0 in (22). Similarly, we have 𝜕^𝑘_𝑥𝐺_𝑡(𝑡)∗𝜕_𝑥1𝜙_𝐿2≤𝐶(1+𝑡)^{−(𝑛/2)((1/𝑝)−(1/2))−((𝑘+1−𝑗)/2)}𝜕^𝑗_𝑥𝜙_𝐿𝑝+𝐶𝑒^−𝑐𝑡𝜕_𝑥^𝑘+𝑙+1𝜙_𝐿2,𝜕^𝑘_𝑥𝐻_𝑡(𝑡)∗𝜓_𝐿2≤𝐶(1+𝑡)^{−(𝑛/2)((1/𝑝)−(1/2))−((𝑘+1−𝑗)/2)}𝜕^𝑗_𝑥𝜓_𝐿𝑝+𝐶𝑒^−𝑐𝑡𝜕_𝑥^𝑘+𝑙+2𝜓_𝐿2 (24)(25) for 0≤𝑗≤𝑘+1.

Proof. We only prove (22). By the Plancherel theorem and (20), we obtain 𝜕^𝑘_𝑥𝐺(𝑡)∗𝜕_𝑥1𝜙²_𝐿2=_|𝜉|≤1𝜉^2𝑘𝐺(𝜉,𝑡)²𝜉₁²𝜙(𝜉)²+𝑑𝜉_|𝜉|≥1𝜉^2𝑘𝐺(𝜉,𝑡)²𝜉₁²𝜙(𝜉)²𝑑𝜉≤𝐶_|𝜉|≤1𝜉^2𝑘𝑒^{−𝑐|𝜉|2𝑡}𝜙(𝜉)²𝑑𝜉+𝐶𝑒^−𝑐𝑡_|𝜉|≥1𝜉^2𝑘+2𝜉²1+𝜉²⁻¹𝜙(𝜉)²𝑑𝜉=∶𝐼₁+𝐼₂. (26) For the term 𝐼₁, letting 1/𝑝^′+1/𝑝=1, we have 𝐼₁≤𝐶_|𝜉|≤1𝜉^2𝑘𝑒^{−𝑐|𝜉|2𝑡}𝜙(𝜉)²𝜉𝑑𝜉≤𝐶^𝑗𝜙²_{𝐿𝑝′|𝜉|≤1}𝜉^{2(𝑘−𝑗)𝑝}𝑒^{−𝑐𝑞|𝜉|2𝑡}𝑑𝜉^1/𝑝≤𝐶(1+𝑡)^{−𝑛((1/𝑝)−(1/2))−(𝑘−𝑗)}𝜕^𝑗_𝑥𝜙²_𝐿𝑝, (27) where we used the Hölder inequality with (2/𝑝^′)+(1/𝑞)=1 and the Hausdorff-Young inequality ‖𝑣‖̂_𝐿𝑝′≤𝐶‖𝑣‖_𝐿^𝑝 for 𝑣=(−Δ)^−(1/2)𝜕^𝑗_𝑥𝜙. On the other hand, we can estimate the term 𝐼₂ simply as 𝐼₂≤𝐶𝑒^−𝑐𝑡_|𝜉|≥1𝜉^2𝑘−2𝜙(𝜉)²𝑑𝜉≤𝐶𝑒^−𝑐𝑡_|𝜉|≥1𝜉^{2(𝑘+𝑙−1)}𝜙(𝜉)²𝑑𝜉≤𝐶𝑒^−𝑐𝑡𝜕_𝑥^{𝑘+𝑙−1}𝜙²_𝐿2, (28) where 𝑘+𝑙−1≥0. Combining (26)–(28) yields (22). We have completed the proof of the Lemma.

Similar to the proof of Lemma 5, it is not difficult to get the following.

Lemma 6. Let 1≤𝑝≤2 and let 𝑘,𝑗,𝑙 be nonnegative integers. Then we have the following estimate: 𝜕^𝑘_𝑥𝐺(𝑡)∗(𝐼−𝑎Δ)⁻¹Δ𝑔_𝐿2≤𝐶(1+𝑡)^{−(𝑛/2)((1/𝑝)−(1/2))−((𝑘+1−𝑗)/2)}𝜕^𝑗_𝑥𝑔_𝐿𝑝+𝐶𝑒^−𝑐𝑡𝜕_𝑥^𝑘+𝑙𝑔_𝐿2 (29) for 0≤𝑗≤𝑘+1. Similarly, we have 𝜕^𝑘_𝑥𝐺_𝑡(𝑡)∗(𝐼−𝑎Δ)⁻¹Δ𝑔_𝐿2≤𝐶(1+𝑡)^{−(𝑛/2)((1/𝑝)−(1/2))−((𝑘+2−𝑗)/2)}𝜕^𝑗_𝑥𝑔_𝐿𝑝+𝐶𝑒^−𝑐𝑡𝜕_𝑥^𝑘+𝑙𝑔_𝐿2 (30) 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 that 𝑓=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∈𝑊^𝑚,𝑞(ℝ^𝑛)⋂𝐿^𝑝(ℝ^𝑛)⋂𝐿^∞(ℝ^𝑛) and ‖𝑣‖_𝐿^∞≤𝜈₀, then the following inequalities hold: 𝜕^𝑚_𝑥𝑓(𝑣)_𝐿𝑟≤𝐶‖𝑣‖_𝐿^𝑝𝜕^𝑚_𝑥𝑣_𝐿𝑞‖𝑣‖_𝐿^𝜃−1∞, (31) where 1/𝑟=(1/𝑝)+(1/𝑞),1≤𝑝,𝑞,𝑟≤+∞.

Proof of Theorem 1. 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: 𝒲(𝑡)=^𝑠+2_𝑘=0sup_{0≤𝜏≤𝑡}(1+𝜏)^𝑘/2𝜕^𝑘_𝑥𝑢(𝜏)_𝐿2. (32) We apply the Gagliardo-Nirenberg inequality. This yields ‖𝑢‖_𝐿^∞≤𝐶𝜕^𝑠0_𝑥𝑢^𝜃_𝐿2‖𝑢‖_𝐿^1−𝜃₂, (33) where 𝑠₀=[𝑛/2]+1 and 𝜃=𝑛/2𝑠₀. It follows from the definition of 𝒲(𝑡) in (32) that ‖𝑢(𝑡)‖_𝐿^∞≤𝐶𝒲(𝑡)(1+𝑡)^−(𝑛/4), (34) provided that 𝑠≥[𝑛/2]−1. Differentiating (18) 𝑘 times with respect to 𝑥 and taking the 𝐿² norm, we obtain 𝜕^𝑘_𝑥𝑢(𝑡)_𝐿2≤𝜕^𝑘_𝑥𝐺(𝑡)∗𝜕_𝑥1𝑣_1𝐿2+𝜕^𝑘_𝑥𝐻(𝑡)∗𝑢_0𝐿2+^𝑡₀𝜕^𝑘_𝑥𝐺(𝑡−𝜏)∗(𝐼−𝑎Δ)⁻¹Δ𝑓(𝑢(𝜏))_𝐿2𝑑𝜏=𝐼₁+𝐼₂+𝐽. (35) Firstly, we estimate 𝐼₁. We get from (22), with 𝑝=2,𝑗=0, and 𝑙=0(𝑙=1 for 𝑘=0), 𝐼₁≤𝐶(1+𝑡)^−(𝑘/2)𝑣_1𝐿2+𝐶𝑒^−𝑐𝑡𝜕^(𝑘−1)+_𝑥𝑣_1𝐿2≤𝐶𝐸₀(1+𝑡)^−(𝑘/2), (36) where (𝑘−1)₊=max{𝑘−1,0}. By using (23) with 𝑝=2, 𝑗=0, and 𝑙=0 to the term 𝐼₂, we obtain 𝐼₂≤𝐶(1+𝑡)^−(𝑘/2)𝑢_0𝐿2+𝐶𝑒^−𝑐𝑡𝜕^𝑘_𝑥𝑢_0𝐿2≤𝐶𝐸₀(1+𝑡)^−(𝑘/2). (37) 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 𝐽₁≤𝐶₀^𝑡/2(1+𝑡−𝜏)^{−((𝑘+1)/2)}𝑓(𝑢)(𝜏)_𝐿2𝑑𝜏+𝐶₀^𝑡/2𝑒^{−𝑐(𝑡−𝜏)}𝜕^𝑘_𝑥𝑓(𝑢)(𝜏)_𝐿2𝑑𝜏. (38) By Lemma 7, we have the estimates ‖𝑓(𝑢)‖_𝐿2≤𝐶‖𝑢‖_𝐿^∞‖𝑢‖_𝐿² and ‖𝜕^𝑘_𝑥𝑓(𝑢)‖_𝐿2≤𝐶‖𝑢‖_𝐿^∞‖𝜕^𝑘_𝑥𝑢‖_𝐿2. Thus by (34), we have 𝑓(𝑢)(𝜏)_𝐿2≤𝐶𝒲(𝑡)²(1+𝜏)^−(𝑛/4),𝜕^𝑘_𝑥𝑓(𝑢)(𝜏)_𝐿2≤𝐶𝒲(𝑡)²(1+𝜏)^{−((𝑛/4)−(𝑘/2))}. (39)(40) Inserting (39) and (40) into (38) yields 𝐽₁≤𝐶𝒲(𝑡)²₀^𝑡/2(1+𝑡−𝜏)^{−((𝑘+1)/2)}(1+𝜏)^−(𝑛/4)(𝑡)𝑑𝜏+𝐶𝒲²₀^𝑡/2𝑒^{−𝑐(𝑡−𝜏)}(1+𝜏)^{−((𝑛/4)−(𝑘/2))}(𝑡)𝑑𝜏≤𝐶𝒲²(1+𝑡)^−(𝑘/2)𝜂(𝑡), (41) 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 𝐽₂≤𝐶^𝑡_𝑡/2(1+𝑡−𝜏)^−(1/2)𝜕^𝑘_𝑥𝑓(𝑢)(𝜏)_𝐿2𝑑𝜏+𝐶^𝑡_𝑡/2𝑒^{−𝑐(𝑡−𝜏)}𝜕^𝑘_𝑥𝑓(𝑢)(𝜏)_𝐿2(𝑡)𝑑𝜏≤𝐶𝒲^2𝑡_𝑡/2(1+𝑡−𝜏)^−(1/2)(1+𝜏)^{−((𝑛/4)−(𝑘/2))}(𝑡)𝑑𝜏≤𝐶𝒲²(1+𝑡)^−(𝑘/2)𝜂(𝑡). (42) Equations (41) and (42) give 𝐽≤𝐶𝒲(𝑡)²(1+𝑡)^{−((𝑛/4)−(𝑘/2))}𝜂(𝑡). (43) Inserting (36), (37), and (43) into (35), we obtain (1+𝑡)^𝑘/2𝜕^𝑘_𝑥𝑢(𝑡)_𝐿2≤𝐶𝐸₀+𝐶𝒲(𝑡)² (44) 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 𝜕^𝑘_𝑥𝑢_𝑡(𝑡)=𝜕^𝑘_𝑥𝐺_𝑡(𝑡)∗𝜕_𝑥1𝑣₁+𝜕^𝑘_𝑥𝐻_𝑡(𝑡)∗𝑢₀+^𝑡₀𝜕^𝑘_𝑥𝐺_𝑡(𝑡−𝜏)∗(𝐼−𝑎Δ)⁻¹Δ𝑓(𝑢)(𝜏)𝑑𝜏. (45) From (45) and Minkowski inequality, we obtain 𝜕^𝑘_𝑥𝑢_𝑡(𝑡)_𝐿2≤𝜕^𝑘_𝑥𝐺_𝑡(𝑡)∗𝜕_𝑥1𝑣_1𝐿2+𝜕^𝑘_𝑥𝐻_𝑡(𝑡)∗𝑢_0𝐿2+^𝑡₀𝜕^𝑘_𝑥𝐺_𝑡(𝑡−𝜏)∗(𝐼−𝑎Δ)⁻¹Δ𝑓(𝑢(𝜏))_𝐿2𝑑𝜏=𝐾₁+𝐾₂+𝐿. (46) It follows from (24) that 𝐾₁≤𝐶(1+𝑡)^{−((𝑘+1)/2)}𝑣_1𝐿2+𝐶𝑒^−𝑐𝑡𝜕_𝑥^𝑘+1𝑣_1𝐿2≤𝐶𝐸₀(1+𝑡)^{−((𝑘+1)/2)}. (47) By using (25), we get 𝐾₂≤𝐶(1+𝑡)^{−((𝑘+1)/2)}𝑢_0𝐿2+𝐶𝑒^−𝑐𝑡𝜕_𝑥^𝑘+2𝑢_0𝐿2≤𝐶𝐸₀(1+𝑡)^{−((𝑘+1)/2)}. (48) 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 𝐿₁≤𝐶₀^𝑡/2(1+𝑡−𝜏)^{−((𝑘+2)/2)}𝑓(𝑢)(𝜏)_𝐿2𝑑𝜏+𝐶₀^𝑡/2𝑒^{−𝑐(𝑡−𝜏)}𝜕^𝑘_𝑥𝑓(𝑢)(𝜏)_𝐿2𝑑𝜏≤𝐶𝒲²(𝑡)₀^𝑡/2(1+𝑡−𝜏)^{−((𝑘+2)/2)}(1+𝜏)^−(𝑛/4)𝑑𝜏+𝐶𝒲²(𝑡)₀^𝑡/2𝑒^{−𝑐(𝑡−𝜏)}(1+𝜏)^{−((𝑛/4)−(𝑘/2))}𝑑𝜏≤𝐶𝒲²(𝑡)(1+𝑡)^{−((𝑘+1)/2)}𝜂(𝑡). (49) Next, for the term 𝐿₁, it follows from (30) with 𝑝=2,  𝑘=0, and 𝑙=0 and (40) that 𝐿₂≤𝐶^𝑡_𝑡/2(1+𝑡−𝜏)⁻¹𝜕^𝑘_𝑥𝑓(𝑢)(𝜏)_𝐿2𝑑𝜏+𝐶^𝑡_𝑡/2𝑒^{−𝑐(𝑡−𝜏)}𝜕^𝑘_𝑥𝑓(𝑢)(𝜏)_𝐿2𝑑𝜏≤𝐶𝒲²(𝑡)^𝑡_𝑡/2(1+𝑡−𝜏)⁻¹(1+𝜏)^{−((𝑛/4)−(𝑘/2))}𝑑𝜏+𝐶𝒲²(𝑡)^𝑡_𝑡/2𝑒^{−𝑐(𝑡−𝜏)}(1+𝜏)^{−((𝑛/4)−(𝑘/2))}𝑑𝜏≤𝐶𝒲²(𝑡)(1+𝑡)^{−((𝑘+1)/2)}𝜂(𝑡). (50) Collecting (46)–(50), which yields 𝜕^𝑘_𝑥𝑢_𝑡(𝑡)_𝐿2≤𝐶𝐸₀(1+𝑡)^{−((𝑘+1)/2)}+𝐶𝒲²(𝑡)(1+𝑡)^{−((𝑘+1)/2)}𝜂(𝑡). (51) 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.

Acknowledgments

This work was supported in part by the NNSF of China (Grant no. 11101144) and Innovation Scientists and the Technicians Troop Construction Projects of Henan Province. Funding scheme for young teachers of Universities of Henan Province.