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

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 dimensions 𝑛≥1. Moreover, when 𝑛≥2, we show that the solution can be approximated by the linear solution as time tends to infinity.

1. Introduction

We investigate the Cauchy problem of the following generalized damped Boussinesq equation: 𝑢_𝑡𝑡−Δ𝑢−2𝑏Δ𝑢_𝑡+𝛼Δ²𝑢=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, 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 𝑢_𝑡𝑡+𝛼Δ²𝑢−Δ𝑢=𝛽Δ𝑢² (3) 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]): 𝑢_𝑡𝑡−Δ𝑢−2𝑏Δ𝑢_𝑡+𝛼Δ²𝑢=𝛽Δ𝑢², (4) 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: 𝜕^𝑘_𝑥𝑢(𝑡)_{𝐻𝑠+2−𝑘}≤𝐶𝐸₀(1+𝑡)^{−𝑛/4−𝑘/2} (5) 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 𝑢_𝐿 to the linearized problem, namely, the problem (1), (2) with 𝑓(𝑢)≡0. More precisely, when 𝑛≥2, we show that 𝜕^𝑘_𝑥𝑢−𝑢_𝐿(𝑡)_{𝐻𝑠+2−𝑘}≤𝐶𝐸²₀(1+𝑡)^{−𝑛/4−𝑘/2}𝜂(𝑡) (6) 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 Section 2, 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 𝑢(𝜉)=ℱ[𝑢]=_ℝ𝑛𝑒^{−𝑖𝜉⋅𝑥}𝑢(𝑥)𝑑𝑥, (7)

ℱ⁻¹

For 1≤𝑝≤∞, 𝐿^𝑝=𝐿^𝑝(ℝ^𝑛) denotes the usual Lebesgue space with the norm ‖⋅‖_𝐿^𝑝. The usual Sobolev space of 𝑠 is defined by 𝑊^𝑠,𝑝=(𝐼−Δ)^−𝑠/2𝐿^𝑝 with the norm ‖𝑓‖_{𝑊𝑠,𝑝}=‖(𝐼−Δ)^𝑠/2𝑓‖_𝐿𝑝; the homogeneous Sobolev space of 𝑠 is defined by 𝑊̇^𝑠,𝑝=(−Δ)^−𝑠/2𝐿^𝑝 with the norm ‖𝑓‖_{̇𝑊𝑠,𝑝}=‖(−Δ)^𝑠/2𝑓‖_𝐿𝑝; especially 𝐻^𝑠=𝑊^𝑠,2, 𝐻̇^𝑠=𝑊̇^𝑠,2. Moreover, we know that 𝑊^𝑠,𝑝=𝐿^𝑝⋂𝑊̇^𝑠,𝑝 for 𝑠≥0.

Finally, in this paper, we denote every positive constant by the same symbol 𝐶 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): 𝑢_𝑡𝑡−Δ𝑢−2𝑏Δ𝑢_𝑡+𝛼Δ²𝑢=0 (8) with the initial data in (2). We apply the Fourier transform to (8). This yields 𝑢_𝑡𝑡+2𝑏|𝜉|²𝑢_𝑡+|𝜉|²+𝛼|𝜉|⁴𝑢=0. (9) The corresponding initial value is given as 𝑡=0∶𝑢=𝑢₀(𝜉),𝑢_𝑡=𝑢₁(𝜉). (10) The characteristic equation of (9) is 𝜆²+2𝑏𝜉²𝜆+𝜉²+𝛼𝜉⁴=0. (11) Let 𝜆=𝜆_±(𝜉) be the corresponding eigenvalues of (11), and we obtain 𝜆_±(𝜉)=−𝑏𝜉²±𝜉𝑏²−𝛼𝜉²−1. (12) The solution to the problem (9), (10) in the Fourier space is then given explicitly in the form 𝑢(𝜉,𝑡)=𝐺(𝜉,𝑡)𝑢₁(𝜉)+𝐻(𝜉,𝑡)𝑢₀(𝜉), (13) 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}. (14)(15) We define 𝐺(𝑥,𝑡) and 𝐻(𝑥,𝑡) by 𝐺(𝑥,𝑡)=ℱ⁻¹𝐺(𝑥)𝐻()𝜉,𝑡)(,𝑥,𝑡=ℱ⁻¹𝐻(𝜉,𝑡)(𝑥), (16)(17) respectively, where ℱ⁻¹ denotes the inverse Fourier transform. Then, applying ℱ⁻¹ to (13), we obtain 𝑢(𝑡)=𝐺(𝑡)∗𝑢₁+𝐻(𝑡)∗𝑢₀. (18) By the Duhamel principle, we obtain the solution formula to (1), (2) as follows: 𝑢(𝑡)=𝐺(𝑡)∗𝑢₁+𝐻(𝑡)∗𝑢₀+^𝑡₀𝐺(𝑡−𝜏)∗Δ𝑓(𝑢)(𝜏)𝑑𝜏. (19)

3. Decay Property

The aim of this section is to establish decay estimates of the solution operators 𝐺(𝑡) and 𝐻(𝑡) appearing in (16) and (17), respectively.

Lemma 1. The solution of the problem (9), (10) satisfies 𝜉²+𝜉⁴𝑢(𝜉,𝑡)²+𝑢_𝑡(𝜉,𝑡)²≤𝐶𝑒^{−𝑐|𝜉|2𝑡}𝜉²+𝜉⁴𝑢₀(𝜉)²+𝑢₁(𝜉)² (20) for 𝜉∈ℝ^𝑛 and 𝑡≥0.

Proof. Multiplying (9) by 𝑢_𝑡 and taking the real part yield 12𝑑|𝑢𝑑𝑡^𝑡|²+|𝜉|²|𝑢|²+𝛼|𝜉|⁴|𝑢|²+2𝑏|𝜉|²|𝑢_𝑡|²=0. (21) Multiplying (9) by 𝑢 and taking the real part, we obtain 12𝑑𝜉𝑑𝑡2𝑏²|𝑢|²+2Re𝑢_𝑡𝑢+|𝜉|²|𝑢|²+𝛼|𝜉|⁴|𝑢|²−|𝑢_𝑡|²=0. (22) Multiplying both sides of (22) by 𝑏|𝜉|² and summing up the resulting equation and (21) yield 𝑑𝑑𝑡𝐸+𝐹=0, (23) where 1𝐸=2|𝑢_𝑡|²+12|𝜉|²|𝑢|²+𝛼2|𝜉|⁴|𝑢|²+𝑏²𝜉⁴|𝑢|²+𝑏𝜉²Re𝑢_𝑡𝑢,𝐹=𝑏|𝜉|²|𝑢_𝑡|²+𝑏|𝜉|⁴|𝑢|²+𝑏𝛼|𝜉|⁶|𝑢|². (24) A simple computation implies that 𝐶𝐸₀≤𝐸≤𝐶𝐸₀, (25) where 𝐸₀=𝜉²+𝜉⁴|𝑢|²+|𝑢_𝑡|². (26) Note that 𝐹≥𝑐|𝜉|²𝐸₀. (27) It follows from (25) that 𝐹≥𝑐|𝜉|²𝐸. (28) Using (23) and (28), we get 𝑑𝑑𝑡𝐸+𝑐|𝜉|²𝐸≤0. (29) Thus 𝐸(𝜉,𝑡)≤𝑒^{−𝑐|𝜉|2𝑡}𝐸(𝜉,0), (30) which together with (25) proves the desired estimates (20). Then we have completed the proof of the lemma.

Lemma 2. Let 𝐺(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 𝜉²+𝜉⁴𝐺(𝜉,𝑡)²+𝐺_𝑡(𝜉,𝑡)²≤𝐶𝑒^{−𝑐|𝜉|2𝑡},|𝜉|²+|𝜉|⁴|𝐻(𝜉,𝑡)|²+|𝐻_𝑡(𝜉,𝑡)|²≤𝐶|𝜉|²+|𝜉|⁴𝑒^{−𝑐|𝜉|2𝑡} (31)(32) for 𝜉∈ℝ^𝑛 and 𝑡≥0.

Proof. Firstly, we investigate the problem (8), (2) with 𝑢₀(𝜉)=0; from (13), we obtain 𝑢(𝜉,𝑡)=𝐺(𝜉,𝑡)𝑢₁(𝜉),𝑢_𝑡(𝜉,𝑡)=𝐺_𝑡(𝜉,𝑡)𝑢₁(𝜉). (33) Substituting the equalities into (20) with 𝑢₀(𝜉)=0, we get (31).
In what follows, we consider the problem (8), (2) with 𝑢₁(𝜉)=0; it follows from (13) that 𝑢(𝜉,𝑡)=𝐻(𝜉,𝑡)𝑢₀(𝜉),𝑢_𝑡(𝜉,𝑡)=𝐻_𝑡(𝜉,𝑡)𝑢₀(𝜉). (34) Substituting the equalities into (20) with 𝑢₁(𝜉)=0, we get the desired estimate (32). The lemma is proved.

Lemma 3. Let 𝐺(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 𝜕^𝑘_𝑥𝐺(𝑡)∗𝜙_𝐿2≤𝐶(1+𝑡)^{−(𝑛/2)(1/𝑝−1/2)−(𝑘−𝑗)/2}𝜕^𝑗_𝑥𝜙_{̇𝑊−1,𝑝}+𝐶𝑒^−𝑐𝑡𝜕_𝑥^{𝑘+𝑙−2}𝜙_𝐿2,𝜕^𝑘_𝑥𝐻(𝑡)∗𝜙_𝐿2≤𝐶(1+𝑡)^{−(𝑛/2)(1/𝑝−1/2)−(𝑘−𝑗)/2}𝜕^𝑗_𝑥𝜙_𝐿𝑝+𝐶𝑒^−𝑐𝑡𝜕_𝑥^𝑘+𝑙𝜙_𝐿2 (35)(36) for 0≤𝑗≤𝑘, where 𝑘+𝑙−2≥0 in (35). Similarly, one has 𝜕^𝑘_𝑥𝐺_𝑡(𝑡)∗𝜙_𝐿2≤𝐶(1+𝑡)^{−(𝑛/2)(1/𝑝−1/2)−(𝑘+1−𝑗)/2}𝜕^𝑗_𝑥𝜙_{̇𝑊−1,𝑝}+𝐶𝑒^−𝑐𝑡𝜕_𝑥^𝑘+𝑙𝜙_𝐿2,𝜕^𝑘_𝑥𝐻_𝑡(𝑡)∗𝜙_𝐿2≤𝐶(1+𝑡)^{−(𝑛/2)(1/𝑝−1/2)−(𝑘+1−𝑗)/2}𝜕^𝑗_𝑥𝜙_𝐿𝑝+𝐶𝑒^−𝑐𝑡𝜕_𝑥^𝑘+𝑙+2𝜙_𝐿2 (37)(38) for 0≤𝑗≤𝑘+1.

Proof. We only prove (35). By the Plancherel theorem, (31), and Hausdorff-Young inequality, we obtain 𝜕^𝑘_𝑥𝐺(𝑡)∗𝜙²_𝐿2=_|𝜉|≤1|𝜉|^2|𝑘||𝐺(𝜉,𝑡)|²|𝜙(𝜉)|²+𝑑𝜉_|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. (39) For the term 𝒜₁, letting 1/𝑝^′+1/𝑝=1, we have𝒜₁≤𝐶_|𝜉|≤1𝜉^2𝑘−2𝑒^{−𝑐|𝜉|2𝑡}𝜙(𝜉)²𝜉𝑑𝜉≤𝐶^𝑗−1𝜙²_{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_{̇𝑊−1,𝑝}, (40) 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 𝒜₂ simply as 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𝜕_𝑥^{𝑘+𝑙−2}𝜙²_𝐿2, (41) where 𝑘+𝑙−2≥0.
Combining the previous three inequalities yields (35). This completes the proof of Lemma 3.

From Lemma 3, we immediately have the following corollary.

Corollary 4. Let 𝐺(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 𝜕^𝑘_𝑥𝐺(𝑡)∗Δ𝑔_𝐿2≤𝐶(1+𝑡)^{−(𝑛/2)(1/𝑝−1/2)−(𝑘+1−𝑗)/2}𝜕^𝑗_𝑥𝑔_𝐿𝑝+𝐶𝑒^−𝑐𝑡𝜕_𝑥^𝑘+𝑙𝑔_𝐿2 (42) for 0≤𝑘≤𝑗+1. Also one has 𝜕^𝑘_𝑥𝐺_𝑡(𝑡)∗Δ𝑔_𝐿2≤𝐶(1+𝑡)^{−(𝑛/2)(1/𝑝−1/2)−(𝑘+2−𝑗)/2}𝜕^𝑗_𝑥𝑔_𝐿𝑝+𝐶𝑒^−𝑐𝑡𝜕_𝑥^𝑘+𝑙+2𝑔_𝐿2 (43) 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 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∞, (44) where 1/𝑟=1/𝑝+1/𝑞, 1≤𝑝,𝑞, 𝑟≤+∞.

Theorem 6. Let 𝑛≥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∈𝐻^𝑠+2⋂𝐿¹, 𝑢₁∈𝐻^𝑠⋂𝑊̇^−1,1. Put 𝐸₀∶=𝑢_0𝐿1+𝑢_{1̇𝑊−1,1}+𝑢_0𝐻𝑠+2+𝑢_1𝐻𝑠. (45) 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 𝑢∈𝐶⁰[0,∞);𝐻^𝑠+2ℝ^𝑛𝐶¹[0,∞);𝐻^𝑠ℝ^𝑛. (46) Moreover, the solution satisfies the decay estimate 𝜕^𝑘_𝑥𝑢(𝑡)_𝐿2≤𝐶𝐸₀(1+𝑡)^{−𝑛/4−𝑘/2},𝜕^𝑙_𝑥𝑢_𝑡(𝑡)_𝐿2≤𝐶𝐸₀(1+𝑡)^{−𝑛/4−(𝑘+1)/2}, (47)(48) 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 𝑀(𝑡)=^𝑠+1_𝑘=0sup_{0≤𝜏≤𝑡}(1+𝜏)^{𝑛/4+𝑘/2}𝜕^𝑘_𝑥𝑢(𝜏)_𝐿2. (49) Here we note that ‖𝑢(𝑡)‖_𝐿^∞≤𝐶𝑀(𝑡)(1+𝑡)^−𝑛/2, (50) provided that 𝑠≥max{0,[𝑛/2]−1}. This follows from the Gagliardo-Nirenberg inequality ‖𝑢‖_𝐿^∞≤𝐶‖𝜕^𝑠0_𝑥𝑢‖^𝜃_𝐿2‖𝑢‖_𝐿^1−𝜃₂ with 𝑠₀=[𝑛/2]+1 and 𝜃=𝑛/2𝑠₀ and the definition of 𝑀(𝑡) in (49).
Applying 𝜕^𝑘_𝑥 to (19) and taking the 𝐿² norm, we obtain 𝜕^𝑘_𝑥𝑢(𝑡)_𝐿2≤𝜕^𝑘_𝑥𝐺(𝑡)∗𝑢_1𝐿2+𝜕^𝑘_𝑥𝐻(𝑡)∗𝑢_0𝐿2+^𝑡₀𝐺(𝑡−𝜏)∗Δ𝑓(𝑢(𝜏))_𝐿2𝑑𝜏∶=𝐼₁+𝐼₂+𝐽. (51) 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 𝐼₁≤𝐶(1+𝑡)^{−𝑛/4−𝑘/2}𝑢_{1̇𝑊−1,1}+𝐶𝑒^−𝑐𝑡𝜕^(𝑘−2)+_𝑥𝑢_1𝐿2≤𝐶𝐸₀(1+𝑡)^{−𝑛/4−𝑘/2}, (52) where (𝑘−2)₊=max{𝑘−2,0}. Similarly, applying (36) with 𝑝=1, 𝑗=0, and 𝑙=0 to the term 𝐼₂, we have 𝐼₂≤𝐶(1+𝑡)^{−𝑛/4−𝑘/2}𝑢_0𝐿1+𝐶𝑒^−𝑐𝑡𝜕^𝑘_𝑥𝑢_0𝐿2≤𝐶𝐸₀(1+𝑡)^{−𝑛/4−𝑘/2}. (53) 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 𝐽₁≤𝐶₀^𝑡/2(1+𝑡−𝜏)^{−𝑛/4−(𝑘+1)/2}𝑓(𝑢)(𝜏)_𝐿1𝑑𝜏+𝐶₀^𝑡/2𝑒^{−𝑐(𝑡−𝜏)}𝜕^𝑘_𝑥𝑓(𝑢)(𝜏)_𝐿2𝑑𝜏∶=𝐽₁₁+𝐽₁₂. (54) Here we see that ‖𝑓(𝑢)‖_𝐿1≤𝐶‖𝑢‖²_𝐿2 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 𝐽₁₁≤𝐶𝑀(𝑡)²₀^𝑡/2(1+𝑡−𝜏)^{−𝑛/4−(𝑘+1)/2}(1+𝜏)^−𝑛/2𝑑𝜏≤𝐶𝑀(𝑡)²(1+𝑡)^{−𝑛/4−(𝑘+1)/2}₀^𝑡/2(1+𝜏)^−𝑛/2𝑑𝜏≤𝐶𝑀(𝑡)²(1+𝑡)^{−𝑛/4−𝑘/2}𝜂(𝑡), (55) where 𝜂(𝑡)=1,𝑛=1,(1+𝑡)^−1/2log(2+𝑡),𝑛=2,(1+𝑡)^−1/2,𝑛≥3. (56) On the other hand, we have ‖𝜕^𝑘_𝑥𝑓(𝑢)‖_𝐿2≤𝐶‖𝑢‖_𝐿^∞‖𝜕^𝑘_𝑥𝑢‖_𝐿2 by Lemma 5. Therefore, using (50), we find that ‖𝜕^𝑘_𝑥𝑓(𝑢)(𝜏)‖_𝐿2≤𝐶𝑀(𝑡)²(1+𝜏)^{−𝑛/4−𝑘/2−𝑛/2}. Consequently, we can estimate the term 𝐽₁₂ as 𝐽₁₂≤𝐶𝑀(𝑡)²₀^𝑡/2𝑒^{−𝑐(𝑡−𝜏)}(1+𝜏)^{−𝑛/4−𝑘/2−𝑛/2}𝑑𝜏≤𝐶𝑀(𝑡)²𝑒^−𝑐𝑡. (57) 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 ‖𝜕^𝑘_𝑥𝑓(𝑢)(𝜏)‖_𝐿2≤𝐶𝑀(𝑡)²(1+𝜏)^{−𝑛/4−𝑘/2−𝑛/2}, we obtain 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)_𝐿2𝑑𝜏≤𝐶𝑀(𝑡)^2𝑡_𝑡/2(1+𝑡−𝜏)^−1/2(1+𝜏)^{−𝑛/4−𝑘/2−𝑛/2}𝑑𝜏≤𝐶𝑀(𝑡)²(1+𝑡)^{−𝑛/4−𝑘/2−(𝑛−1)/2}. (58) Thus we have shown that 𝐽≤𝐶𝑀(𝑡)²(1+𝑡)^{−𝑛/4−𝑘/2}𝜂(𝑡). (59) Substituting all these estimates into (51), we obtain (1+𝑡)^{𝑛/4+𝑘/2}𝜕^𝑘_𝑥𝑢(𝑡)_𝐿2≤𝐶𝐸₀+𝐶𝑀(𝑡)² (60) 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 𝑢_𝑡. For this purpose, we differentiate (19) with respect to 𝑡 to obtain 𝑢_𝑡(𝑡)=𝐺_𝑡(𝑡)∗𝑢₁+𝐻_𝑡(𝑡)∗𝑢₁+^𝑡₀𝐺_𝑡(𝑡−𝜏)∗Δ𝑓(𝑢)(𝜏)𝑑𝜏. (61) Applying 𝜕^𝑘_𝑥 to (61) and taking the 𝐿² norm, we have 𝜕^𝑘_𝑥𝑢_𝑡(𝑡)_𝐿2≤𝜕^𝑘_𝑥𝐺_𝑡(𝑡)∗𝑢_1𝐿2+𝜕^𝑘_𝑥𝐻_𝑡(𝑡)∗𝑢_0𝐿2+^𝑡₀𝜕^𝑘_𝑥𝐺_𝑡(𝑡−𝜏)∗Δ𝑓(𝑢)(𝜏)_𝐿2𝑑𝜏∶=𝐼^′₁+𝐼^′₂+𝐽^′, (62) where 0≤𝑘≤𝑠. For the term 𝐼^′₁, we apply (37) with 𝑝=1, 𝑗=0, and 𝑙=0 to get 𝐼^′₁≤𝐶(1+𝑡)^{−𝑛/4−(𝑘+1)/2}𝑢_{1̇𝑊−1,1}+𝑒^−𝑐𝑡𝜕^𝑘_𝑥𝑢_1𝐿2≤𝐶𝐸₀(1+𝑡)^{−𝑛/4−(𝑘+1)/2}. (63) Also, for the term 𝐼^′₂, applying (38) with 𝑝=1, 𝑗=0, and 𝑙=0, we have 𝐼^′₂≤𝐶(1+𝑡)^{−𝑛/4−(𝑘+1)/2}𝑢_0𝐿1+𝑒^−𝑐𝑡𝜕_𝑥^𝑘+2𝑢_0𝐿2≤𝐶𝐸₀(1+𝑡)^{−𝑛/4−(𝑘+1)/2}. (64) To estimate the nonlinear term 𝐽^′, we divide 𝐽^′₁ as 𝐽^′=𝐽^′₁+𝐽^′₂, where 𝐽^′₁ and 𝐽^′₂ correspond to the time intervals [0,𝑡/2] and [𝑡/2,𝑡], respectively. For the term 𝐽^′₁, we apply (43) with 𝑝=1, 𝑗=0, and 𝑙=0. This yields 𝐽^′₁≤𝐶₀^𝑡/2(1+𝑡−𝜏)^{−𝑛/4−(𝑘+2)/2}𝑓(𝑢)(𝜏)_𝐿1𝑑𝜏+𝐶₀^𝑡/2𝑒^{−𝑐(𝑡−𝜏)}𝜕_𝑥^𝑘+2𝑓(𝑢)(𝜏)_𝐿2𝑑𝜏∶=𝐽^′₁₁+𝐽^′₁₂. (65) Since ‖𝑓(𝑢)(𝜏)‖_𝐿1≤𝐶𝑀(𝑡)²(1+𝜏)^−𝑛/2 as before, we can estimate the term 𝐽^′₁₁ as 𝐽^′₁₁≤𝐶𝑀(𝑡)²₀^𝑡/2(1+𝑡−𝜏)^{−𝑛/4−(𝑘+2)/2}(1+𝜏)^−𝑛/2(𝑡)𝑑𝜏≤𝐶𝑀²(1+𝑡)^{−𝑛/4−(𝑘+1)/2}𝜂(𝑡). (66) Also, the term 𝐽^′₁₂ is estimated similarly as before, and we have 𝐽^′₁₂≤𝐶𝑀(𝑡)²𝑒^−𝑐𝑡. Finally, we estimate the term 𝐽^′₂ by applying (43) with 𝑝=2, 𝑗=𝑘+2, and 𝑙=0 and obtain 𝐽^′₂≤𝐶^𝑡_𝑡/2𝜕_𝑥^𝑘+2𝑓(𝑢)(𝜏)_𝐿2𝑑𝜏+𝐶^𝑡_𝑡/2𝑒^{−𝑐(𝑡−𝜏)}𝜕_𝑥^𝑘+2𝑓(𝑢)(𝜏)_𝐿2𝑑𝜏≤𝐶𝑀(𝑡)^2𝑡_𝑡/2(1+𝜏)^{−𝑛/4−(𝑘+2)/2−𝑛/2}𝑑𝜏≤𝐶𝑀(𝑡)²(1+𝑡)^{−𝑛/4−(𝑘+1)/2−(𝑛−1)/2}, (67) where we used the estimate ‖𝜕_𝑥^𝑘+2𝑓(𝑢)(𝜏)‖_𝐿2≤𝐶𝑀(𝑡)²(1+𝜏)^{−𝑛/4−(𝑘+2)/2−𝑛/2}. Consequently we have shown that 𝐽^′≤𝐶𝑀(𝑡)²(1+𝑡)^{−𝑛/4−(𝑘+1)/2}𝜂(𝑡). (68) 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 Theorem 6 shows that when 𝑛≥2, the solution 𝑢 to the integral equation (19) is asymptotic to the linear solution 𝑢_𝐿 given by the formula 𝑢_𝐿(𝑡)∶=𝐺(𝑡)∗𝑢₁+𝐻(𝑡)∗𝑢₀ in (18) as 𝑡→∞. This result is stated as follows.

Corollary 7. Let 𝑛≥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 𝑢_𝐿 to the linearized problem (8), (2) as 𝑡→∞. More precisely, one has the following asymptotic relations: 𝜕^𝑘_𝑥(𝑢−𝑢_𝐿)(𝑡)_𝐿2≤𝐶𝐸²₀(1+𝑡)^{−𝑛/4−𝑘/2}𝜂(𝑡),𝜕^𝑘_𝑥(𝑢−𝑢_𝐿)_𝑡(𝑡)_𝐿2≤𝐶𝐸²₀(1+𝑡)^{−𝑛/4−(𝑘+1)/2}𝜂(𝑡) (69) for 0≤𝑘≤𝑠+2 and 0≤𝑘≤𝑠, respectively, where 𝑢_𝐿(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 solution 𝑢 to the problem (1), (2) can be approximated by the linear solution 𝑢_𝐿. The aim of this section is to derive a simpler asymptotic profile of the linear solution 𝑢_𝐿.

In Section 2, we derive the solution formula 𝑢_𝐿(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)𝑢₁(𝜉)+𝐻(𝜉,𝑡)𝑢₀(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 𝜆_±(𝜉)=−𝑏𝜉²±𝑖𝜉1+𝛼−𝑏²𝜉²=−𝑏𝜉²𝜉1±𝑖1+2𝛼−𝑏²𝜉²+𝑂𝜉⁴=±𝑖𝜉−𝑏𝜉²+𝑂𝜉³,1𝜆₊−𝜆₋=12𝑖𝜉1+𝛼−𝑏²𝜉²=1𝜉12𝑖1−2𝛼−𝑏²𝜉²+𝑂𝜉⁴. (70) Inserting the previous equalities into (14) and (15), respectively, we arrive at 𝐺()=𝑒𝜉,𝑡^𝜆+𝑡−𝑒^𝜆−𝑡𝜆₊−𝜆₋=1𝜉𝑒2𝑖^{−𝑏|𝜉|2𝑡}𝑒^{|𝜉|𝑡𝑖}−𝑒^{−|𝜉|𝑡𝑖}+𝑒^{−𝑏|𝜉|2𝑡}𝑂𝜉+𝑂𝜉³𝑡,𝐻()=𝜆𝜉,𝑡⁺𝑒^𝜆−𝑡−𝜆₋𝑒^𝜆+𝑡𝜆₊−𝜆₋=12𝑒^{−𝑏|𝜉|2𝑡}𝑒^{|𝜉|𝑡𝑖}+𝑒^{−|𝜉|𝑡𝑖}+𝑒^{−𝑏|𝜉|2𝑡}𝑂𝜉+𝑂𝜉³𝑡 (71) for 𝜉→0. Taking 𝐺₀()=1𝜉,𝑡𝜉𝑒2𝑖^{−𝑏|𝜉|2𝑡}𝑒^{𝑖|𝜉|𝑡}−𝑒^{−𝑖|𝜉|𝑡},𝐻₀()=1𝜉,𝑡2𝑒^{−𝑏|𝜉|2𝑡}𝑒^{𝑖|𝜉|𝑡}+𝑒^{−𝑖|𝜉|𝑡}. (72) Then we find that 𝐺−𝐺₀(𝜉,𝑡)≤𝐶𝑒^{−𝑐|𝜉|2𝑡},𝐻−𝐻₀(𝜉,𝑡)≤𝐶𝜉𝑒^{−𝑐|𝜉|2𝑡} (73) for |𝜉|≤𝑟₀, where 𝑟₀ is a small positive constant.

We now define 𝑢_𝐿 by 𝑢_𝐿(𝑡)=𝐺₀(𝑡)∗𝑢₁+𝐻₀(𝑡)∗𝑢₀. (74) Then 𝑢_𝐿 gives a asymptotic profile of the linear solution 𝑢_𝐿. In fact we have the following.

Theorem 8 (𝑛≥1 and 𝑠≥0). Assume that 𝑢₀∈𝐻^𝑠+2∩𝐿¹ and 𝑢₁∈𝐻^𝑠∩𝑊̇^−1,1, and put 𝐸₀=‖𝑢₀‖_𝐿1+‖𝑢₁‖_̇𝑊−1,1+‖𝑢₀‖_𝐻𝑠+2+‖𝑢₁‖_𝐻𝑠. Let 𝑢_𝐿 be the linear solution, and let 𝑢_𝐿 be defined by (74). Then one has 𝜕^𝑘_𝑥(𝑢_𝐿−𝑢_𝐿)(𝑡)_𝐿2≤𝐶𝐸₀(1+𝑡)^{−𝑛/4−(𝑘+1)/2} (75) for 0≤𝑘≤𝑠+2.

Proof. Note that (𝑢_𝐿−𝑢_𝐿)(𝑡)=(𝐺−𝐺₀)(𝑡)∗𝑢₁+(𝐻−𝐻₀)(𝑡)∗𝑢₀, so for the proof of (75), it suffices to show the following estimates: 𝜕^𝑘_𝑥(𝐺−𝐺₀)(𝑡)∗𝑢_1𝐿2≤𝐶(1+𝑡)^{−(𝑛/2)(1/𝑝−1/2)−(𝑘+1−𝑗)/2}𝜕^𝑗_𝑥𝑢_{1̇𝑊−1,𝑝}+𝐶𝑒^−𝑐𝑡𝜕_𝑥^{𝑘+𝑙−2}𝑢_1𝐿2,𝜕^𝑘_𝑥(𝐻−𝐻₀)(𝑡)∗𝑢_0𝐿2≤𝐶(1+𝑡)^{−(𝑛/2)(1/𝑝−1/2)−(𝑘+1−𝑗)/2}𝜕^𝑗_𝑥𝑢_0𝐿𝑝+𝐶𝑒^−𝑐𝑡𝜕_𝑥^𝑘+𝑙𝑢_0𝐿2, (76) 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.

Acknowledgment

This work was supported in part by the NNSF of China (Grant no. 11101144).