Asymptotic Behavior of Solutions to the Damped Nonlinear Hyperbolic Equation

Abstract

We consider the Cauchy problem for the damped nonlinear hyperbolic equation in n-dimensional space. Under small condition on the initial value, the global existence and asymptotic behavior of the solution in the corresponding Sobolev spaces are obtained by the contraction mapping principle.

1. Introduction

We investigate the Cauchy problem for the following damped nonlinear hyperbolic equation: 𝑢_𝑡𝑡+𝑘₁Δ²𝑢+𝑘₂Δ²𝑢_𝑡=Δ𝑓(Δ𝑢) (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, 𝑘₁>0 and 𝑘₂>0 are constants. The nonlinear term 𝑓(𝑢)=𝑂(𝑢^1+𝜃) and 𝜃 is a positive integer.

Equation (1) is a model in variational form for the neo-Hookean elastomer rod and describes the motion of a neo-Hookean elastomer rod with internal damping; for more detailed physical background, we refer to [1]. In [1], the authors have studied a general class of abstract evolution equations 𝑤_𝑡𝑡+𝐴₁𝑤+𝐴₂𝑤_𝑡+𝑁^∗𝑔(𝑁𝑤)=𝑓, (3) where 𝐴₁, 𝐴₂, 𝑁, and 𝑓 satisfy certain assumptions. For quite general conditions on the nonlinear term, global existence, uniqueness, regularity, and continuous dependence on the initial value of a generalized solution to (3) in a bounded domain of ℝ^𝑛 were obtained. Equation (1) fits the abstract framework of [1]. The local well-posedness for the Cauchy problem for (1), (2) in three-dimensional space was obtained by Chen and Da [2]. More precisely, they proved local existence and uniqueness of weak solutions to (1), (2) under the assumption that 𝑢₀∈𝐻⁶(ℝ³), 𝑢₁∈𝐻⁴(ℝ³). Local existence and uniqueness of classical solutions to (1), (2) were also established, provided that 𝑢₀∈𝐻¹²(ℝ³), 𝑢₁∈𝐻¹⁰(ℝ³). Their method is to first establish local-in-time well-posedness of a periodic version of (1), (2) and then construc a solution to (1), (2) as a limit of periodic solutions with divergent periods. This paper also arrived at some sufficient conditions for blow-up of the solution in finite time, and an example was given. Song and Yang [3] studied the existence and nonexistence of global solutions to the Cauchy problem for (1) in one-dimensional space. The boundary value problems for (1) are investigated (see [4, 5]). Equation (1) is a fifth-order wave equation. For more higher order wave equations, we refer to [6–8] and references therein.

The main purpose of this paper is to establish global existence and asymptotic behavior of solutions to (1), (2) by using the contraction mapping principle. Firstly, we consider the decay property of the following linear equation: 𝑢_𝑡𝑡+𝑘₁Δ²𝑢+𝑘₂Δ²𝑢_𝑡=0. (4) We obtain the following decay estimate of solutions to (4), (2) 𝜕^𝑘_𝑥𝑢(𝑡)_𝐿2≤𝐶(1+𝑡)^{−(𝑛/8+𝑘/4)}×𝑢_0𝐿1+𝑢_1̇𝐻1−2+𝑢_0𝐻𝑠+4+𝑢_1𝐻𝑠(),𝜕for𝑘≤𝑠+4_𝑙𝑥𝑢_𝑡(𝑡)_𝐿2≤𝐶(1+𝑡)^{−(𝑛/8+𝑙/4+1/2)}×𝑢_0𝐿1+𝑢_1̇𝐻1−2+𝑢_0𝐻𝑠+4+𝑢_1𝐻𝑠for(𝑙≤𝑠). (5) Based on the above estimates, we define a solution space with time weighted norms, and then global existence and asymptotic behavior of solutions to (1), (2) are obtained by using the contraction mapping principle. More precisely, we prove global existence and the following decay estimate of solution to (1), (2): 𝜕^𝑘_𝑥𝑢(𝑡)_𝐿2≤𝐶𝐸₀(1+𝑡)^{−(𝑛/8+𝑘/4)},𝜕^𝑙_𝑥𝑢_𝑡(𝑡)_𝐿2≤𝐶𝐸₀(1+𝑡)^{−(𝑛/8+𝑙/4+1/2)} (6) for 𝑘≤𝑠+4, 𝑙≤𝑠, and 𝑠≥[𝑛/2]+1. Here 𝑢₀∈𝐻^𝑠+4(ℝ^𝑛)⋂𝐿¹(ℝ^𝑛), 𝑢₁∈𝐻^𝑠(ℝ^𝑛)⋂𝐻̇₁⁻²(ℝ^𝑛), and 𝐸₀=‖𝑢₀‖_𝐿1+‖𝑢₁‖_̇𝐻1−2+‖𝑢₀‖_𝐻𝑠+4+‖𝑢₁‖_𝐻𝑠 is assumed to be suitably small. When 𝑛=3, our result allows for the initial data 𝑢₀∈𝐻⁶(ℝ³), 𝑢₁∈𝐻²(ℝ³). But in [2], the authors proved local existence and uniqueness of weak solutions to (1), (2) under the assumption that 𝑢₀∈𝐻⁶(ℝ³), 𝑢₁∈𝐻⁴(ℝ³), so our result improves the regularity of the initial condition for the time derivative. This improvement is due to the strong damping term Δ²𝑢_𝑡 since the strong damping term Δ²𝑢_𝑡 has stronger dissipative effect than the damping 𝑢_𝑡. The stronger dissipative effect has been exhibited in the study of the strongly damped wave equation and related problems; see, for instance, [9].

The global existence and asymptotic behavior of solutions to hyperbolic-type equations have been investigated by many authors. We refer to [10–15] for hyperbolic equations, [16–21] for damped wave equation, and [22, 23] for various aspects of dissipation of the plate equation.

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) and we denote its inverse transform by 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 𝐻^𝑠_𝑝=(𝐼−Δ)^−𝑠/2𝐿^𝑝 with the norm ‖𝑓‖_𝐻𝑠𝑝=‖(𝐼−Δ)^𝑠/2𝑓‖_𝐿𝑝; the homogeneous Sobolev space of 𝑠 is defined by 𝐻̇^𝑠_𝑝=(−Δ)^−𝑠/2𝐿^𝑝 with the norm ‖𝑓‖_𝐻𝑠𝑝=‖(−Δ)^𝑠/2𝑓‖_𝐿𝑝; especially 𝐻^𝑠=𝐻^𝑠₂, 𝐻̇^𝑠=𝐻̇^𝑠₂. Moreover, we know that 𝐻^𝑠_𝑝=𝐿^𝑝⋂𝐻̇^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.

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 discuss the linear problem and show the decay estimates. Finally, we prove global existence and asymptotic behavior of solutions for the Cauchy problem (1), (2) in Section 5.

2. Solution Formula

The aim of this section is to derive the solution formula for the problem (1), (2). We first investigate (4). Taking the Fourier transform, we have 𝑢_𝑡𝑡+𝑘₂𝜉⁴𝑢_𝑡+𝑘₁𝜉⁴𝑢=0. (8) The corresponding initial value is given as 𝑡=0∶𝑢=𝑢₀(𝜉),𝑢_𝑡=𝑢₁(𝜉). (9) The characteristic equation of (8) is 𝜆²+𝑘₂𝜉⁴𝜆+𝑘₁𝜉⁴=0. (10) Let 𝜆=𝜆_±(𝜉) be the corresponding eigenvalues of (10), and we obtain 𝜆_±(𝜉)=−𝑘₂𝜉⁴±𝜉²𝑘²₂𝜉⁴−4𝑘₁2. (11) The solution to the problem (8)-(9) is given in the form 𝑢(𝜉,𝑡)=𝐺(𝜉,𝑡)𝑢₁(𝜉)+𝐻(𝜉,𝑡)𝑢₀(𝜉), (12) 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}. (13)(14) 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 (12), we obtain 𝑢(𝑡)=𝐺(𝑡)∗𝑢₁+𝐻(𝑡)∗𝑢₀. (15) By the Duhamel principle, we obtain the solution formula to (1), (2) 𝑢(𝑡)=𝐺(𝑡)∗𝑢₁+𝐻(𝑡)∗𝑢₀+^𝑡₀𝐺(𝑡−𝜏)∗Δ𝑓(Δ𝑢)(𝜏)𝑑𝜏. (16)

3. Decay Property

The aim of this section is to establish decay estimates of the solution operators 𝐺(𝑡) and 𝐻(𝑡) appearing in the solution formula (15).

Lemma 1. The solution of the problem (8), (9) satisfies 𝜉⁴+𝜉⁸𝑢(𝜉,𝑡)²+𝑢_𝑡(𝜉,𝑡)²≤𝐶𝑒^{−𝑐𝜔(𝜉)𝑡}𝜉⁴+𝜉⁸𝑢₀(𝜉)²+𝑢₁(𝜉)² (17) for 𝜉∈ℝ^𝑛 and 𝑡≥0, where 𝜔(𝜉)=𝜉⁴,𝜉≤𝑅₀1,𝜉≥𝑅₀. (18)

Proof. Multiplying (8) by 𝑢_𝑡 and taking the real part yield 12𝑑𝑢𝑑𝑡^𝑡2+𝑘₂𝜉⁴|𝑢|²+𝑘₁𝜉⁴𝑢_𝑡²=0. (19) Multiplying (8) by 𝑢 and taking the real part, we obtain 12𝑑𝑘𝑑𝑡¹𝜉⁴|𝑢|²+2Re𝑢_𝑡𝑢+𝑘₂𝜉⁴|𝑢|²−𝑢_𝑡²=0. (20) Multiplying both sides of (19) and (20) by 2 and 𝑘₁|𝜉|⁴ and summing up the resulting equation yield 𝑑𝑑𝑡𝐸+𝐹=0, (21) where 𝐸=𝑢_𝑡²+𝑘₂𝜉⁴|𝑢|²+12𝜉⁸|𝑢|²+𝑘₁𝜉⁴Re𝑢_𝑡𝑢,𝐹=𝑘₁𝑘₂𝜉⁸|𝑢|²+𝑘₁𝜉⁴𝑢_𝑡². (22) A simple computation implies that 𝐶𝐸₀≤𝐸≤𝐶𝐸₀, (23) where 𝐸₀=𝜉⁴+𝜉⁸|𝑢|²+𝑢_𝑡². (24) Note that 𝑐𝜉𝐹≥⁴𝜉⁴+𝜉⁸|𝑢|²+𝑢_𝑡²,𝜉≤𝑅₀𝑐𝜉⁴+𝜉⁸|𝑢|²+𝑢_𝑡²,𝜉≥𝑅₀. (25) It follows from (23) that 𝐹≥𝑐𝜔(𝜉)𝐸. (26) Using (21) and (26), we get 𝑑(𝜉)𝑑𝑡𝐸+𝑐𝜔𝐸≤0. (27) Thus 𝐸(𝜉,𝑡)≤𝑒^{−𝑐𝜔(𝜉)𝑡}𝐸(𝜉,0), (28) which together with (23) proves the desired estimates (17). Then we have completed the proof of the lemma.

Lemma 2. Let 𝐺(𝜉,𝑡) and 𝐻(𝜉,𝑡) be the fundamental solution of (4) in the Fourier space, which are given in (13) and (14), respectively. Then one has the estimates 𝜉⁴+𝜉⁸𝐺(𝜉,𝑡)²+𝐺_𝑡(𝜉,𝑡)²≤𝐶𝑒^{−𝑐𝜔(𝜉)𝑡},𝜉⁴+𝜉⁸𝐻(𝜉,𝑡)²+𝐻_𝑡(𝜉,𝑡)²≤𝐶𝜉⁴+𝜉⁸𝑒^{−𝑐𝜔(𝜉)𝑡} (29)(30) for 𝜉∈ℝ^𝑛 and 𝑡≥0, where 𝜔(𝜉)=𝜉⁴,𝜉≤𝑅₀1,𝜉≥𝑅₀. (31)

Proof. If 𝑢₀(𝜉)=0, from (12), we obtain 𝑢(𝜉,𝑡)=𝐺(𝜉,𝑡)𝑢₁(𝜉),𝑢_𝑡(𝜉,𝑡)=𝐺_𝑡(𝜉,𝑡)𝑢₁(𝜉). (32) Substituting the equalities into (17) with 𝑢₀(𝜉)=0, we get (29).
In what follows, we consider 𝑢₁(𝜉)=0, and it follows from (12) that 𝑢(𝜉,𝑡)=𝐻(𝜉,𝑡)𝑢₀(𝜉),𝑢_𝑡(𝜉,𝑡)=𝐻_𝑡(𝜉,𝑡)𝑢₀(𝜉). (33) Substituting the equalities into (17) with 𝑢₁(𝜉)=0, we get the desired estimate (30). The lemma is proved.

Lemma 3. Let 𝑘 and 𝑗 be nonnegative integer. Then one has 𝜕^𝑘_𝑥𝐺(𝑡)∗𝜙_𝐿2≤𝐶(1+𝑡)^{−(𝑛/8+𝑘/4+𝑗/4−1/2)}×𝜙_̇𝐻1^−𝑗+𝐶𝑒^−𝑐𝑡𝜕^(𝑘−4)+_𝑥𝜙_𝐿2,𝜕^𝑘_𝑥𝐻(𝑡)∗𝜙_𝐿2≤𝐶(1+𝑡)^{−(𝑛/8+𝑘/4+𝑗/4)}×𝜙_̇𝐻1^−𝑗+𝐶𝑒^−𝑐𝑡𝜕^𝑘_𝑥𝜙_𝐿2,𝜕^𝑘_𝑥𝐺_𝑡(𝑡)∗𝜙_𝐿2≤𝐶(1+𝑡)^{−(𝑛/8+𝑘/4+𝑗/4)}×𝜙_̇𝐻1^−𝑗+𝐶𝑒^−𝑐𝑡𝜕^𝑘_𝑥𝜙_𝐿2,𝜕^𝑘_𝑥𝐻_𝑡(𝑡)∗𝜙_𝐿2≤𝐶(1+𝑡)^{−(𝑛/8+𝑘/4+𝑗/4+1/2)}×𝜙_̇𝐻1^−𝑗+𝐶𝑒^−𝑐𝑡𝜕_𝑥^(𝑘+4)𝜙_𝐿2,𝜕^𝑘_𝑥𝐺(𝑡)∗Δ𝑔_𝐿2≤𝐶(1+𝑡)^{−(𝑛/8+𝑘/4)}×𝑔_𝐿1+𝐶𝑒^−𝑐𝑡𝜕^(𝑘−2)+_𝑥𝑔_𝐿2,𝜕^𝑘_𝑥𝐺_𝑡(𝑡)∗Δ𝑔_𝐿2≤𝐶(1+𝑡)^{−(𝑛/8+𝑘/4+1/2)}×𝑔_𝐿1+𝐶𝑒^−𝑐𝑡𝜕_𝑥^(𝑘+2)𝑔_𝐿2. (34)(35)(36)(37)(38)(39) Here (𝑘−4)₊=max{0,𝑘−4} in (34) and (𝑘−2)₊=max{0,𝑘−2} in (38).

Proof. By the Plancherel theorem and (29), Hausdorff-Young inequality, we obtain 𝜕^𝑘_𝑥𝐺(𝑡)∗𝜙²_𝐿2=_{|𝜉|≤𝑅0}𝜉^2|𝑘|𝐺(𝜉,𝑡)²𝜙(𝜉)²+𝑑𝜉_{|𝜉|≥𝑅0}𝜉^2|𝑘|𝐺(𝜉,𝑡)²𝜙(𝜉)²≤𝑑𝜉_{|𝜉|≤𝑅0}𝜉^2|𝑘|−4𝑒^{−𝑐|𝜉|4𝑡}𝜙(𝜉)²𝑑𝜉+𝐶𝑒^−𝑐𝑡_{|𝜉|≥𝑅0}𝜉^2𝑘𝜉⁸+𝜉⁴⁻¹𝜙(𝜉)²≤𝑑𝜉_{|𝜉|≤𝑅0}𝜉^{2𝑘−4+2𝑗}𝑒^{−𝑐|𝜉|4𝑡}𝜉^−2𝑗𝜙(𝜉)²𝑑𝜉+𝐶𝑒^−𝑐𝑡𝜕^(𝑘−4)+_𝑥𝜙²_𝐿2≤𝐶𝜉^−𝑗𝜙(𝜉)²_{𝐿∞|𝜉|≤𝑅0}𝜉^{2𝑘−4+2𝑗}𝑒^{−𝑐|𝜉|4𝑡}𝑑𝜉+𝐶𝑒^−𝑐𝑡𝜕^(𝑘−4)+_𝑥𝜙²_𝐿2≤𝐶(1+𝑡)^{−(𝑛/4+𝑘/2+𝑗/2−1)}(−Δ)^−𝑗/2𝜙²_𝐿1+𝐶𝑒^−𝑐𝑡𝜕^(𝑘−4)+_𝑥𝜙²_𝐿2. (40) Here (𝑘−4)₊=max{0,𝑘−4} and 𝑅₀ is a small positive constant in Lemma 1. Thus (34) follows.
Similarly, using (29) and (30), respectively, we can prove (35)–(37).
In what follows, we prove (38). By the Plancherel theorem, (29), and Hausdorff-Young inequality, we have 𝜕^𝑘_𝑥𝐺(𝑡)∗Δ𝑔²_𝐿2=_{|𝜉|≤𝑅0}𝜉^2|𝑘|𝐺(𝜉,𝑡)²𝜉⁴𝑔(𝜉)²+𝑑𝜉_{|𝜉|≥𝑅0}𝜉^2𝑘𝐺(𝜉,𝑡)²𝜉⁴𝑔(𝜉)²≤𝑑𝜉_{|𝜉|≤𝑅0}𝜉^2𝑘𝑒^{−𝑐|𝜉|4𝑡}𝑔(𝜉)²𝑑𝜉+𝐶𝑒^−𝑐𝑡_{|𝜉|≥𝑅0}𝜉^2𝑘1+𝜉⁴⁻¹𝑔(𝜉)²𝑔(𝜉𝑑𝜉≤𝐶)²_{𝐿∞|𝜉|≤𝑅0}𝜉^2𝑘𝑒^{−𝑐|𝜉|4𝑡}𝑑𝜉+𝐶𝑒^−𝑐𝑡𝜕^(𝑘−2)+_𝑥𝑔²_𝐿2≤𝐶(1+𝑡)^{−(𝑛/4+𝑘/2)}𝑔²_𝐿1+𝐶𝑒^−𝑐𝑡𝜕^(𝑘−2)+_𝑥𝑔²_𝐿2, (41) where 𝑅₀ is a small positive constant in Lemma 1. Thus (38) follows. Similarly, we can prove (39). Thus we have completed the proof of the lemma.

4. Decay Estimate of Solutions to (4), (2)

Theorem 4. Assume that 𝑢₀∈𝐻^𝑠+4(ℝ^𝑛)⋂𝐿¹(ℝ^𝑛), 𝑢₁∈𝐻^𝑠(ℝ^𝑛)⋂𝐻̇₁⁻²(ℝ^𝑛)(𝑠≥[𝑛/2]+1). Then the classical solution 𝑢(𝑥,𝑡) to (4), (2), which is given by the formula (15), satisfies the decay estimate 𝜕^𝑘_𝑥𝑢(𝑡)_𝐿2≤𝐶(1+𝑡)^{−(𝑛/8+𝑘/4)}×𝑢_0𝐿1+𝑢_1̇𝐻1−2+𝑢_0𝐻𝑠+4+𝑢_1𝐻𝑠(),𝜕𝑓𝑜𝑟𝑘≤𝑠+4_𝑙𝑥𝑢_𝑡(𝑡)_𝐿2≤𝐶(1+𝑡)^{−(𝑛/8+𝑙/4+1/2)}×𝑢_0𝐿1+𝑢_1̇𝐻1−2+𝑢_0𝐻𝑠+4+𝑢_1𝐻𝑠(),𝜕𝑓𝑜𝑟𝑙≤𝑠_ℎ𝑥𝑢(𝑡)_𝐿∞≤𝐶(1+𝑡)^{−(𝑛/4+ℎ/4)}×𝑢_0𝐿1+𝑢_1̇𝐻1−2+𝑢_0𝐻𝑠+4+𝑢_1𝐻𝑠𝑛𝑓𝑜𝑟ℎ≤𝑠−2+3. (42)(43)(44)

Proof. Firstly, we prove (42). Using (34) and (35), for 𝑘≤𝑠+4, we obtain 𝜕^𝑘_𝑥𝑢(𝑡)_𝐿2≤𝜕^𝑘_𝑥𝐺(𝑡)∗𝑢_1𝐿2+𝐶𝜕^𝑘_𝑥𝐻(𝑡)∗𝑢_0𝐿2≤𝐶(1+𝑡)^{−(𝑛/8+𝑘/4)}𝑢_0𝐿1+𝑢_1̇𝐻1−2+𝐶𝑒^−𝑐𝑡𝑢_0𝐻𝑠+4+𝑢_1𝐻𝑠≤𝐶(1+𝑡)^{−(𝑛/8+𝑘/4)}×𝑢_0𝐿1+𝑢_1̇𝐻1−2+𝑢_0𝐻𝑠+4+𝑢_1𝐻𝑠. (45) For 𝑙≤𝑠, it follows from (36) and (37) that 𝜕^𝑙_𝑥𝑢_𝑡(𝑡)_𝐿2≤𝜕^𝑙_𝑥𝐺_𝑡(𝑡)∗𝑢_1𝐿2+𝐶𝜕^𝑙_𝑥𝐻_𝑡(𝑡)∗𝑢_0𝐿2≤𝐶(1+𝑡)^{−(𝑛/8+𝑙/4+1/2)}𝑢_0𝐿1+𝑢_1̇𝐻1−2+𝐶𝑒^−𝑐𝑡𝑢_0𝐻𝑠+4+𝑢_1𝐻𝑠≤𝐶(1+𝑡)^{−(𝑛/8+𝑙/4+1/2)}×𝑢_0𝐿1+𝑢_1̇𝐻1−2+𝑢_0𝐻𝑠+4+𝑢_1𝐻𝑠. (46) Equation (44) follows from (42) and Gagliardo-Nirenberg inequality. The lemma is proved.

5. Global Existence and Asymptotic Behavior

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 comes from [24] (see also [25]).

Lemma 5. Let 𝑠 and 𝜃 be positive integers, 𝛿>0, 𝑝,  𝑞, 𝑟∈[1,∞] satisfy 1/𝑟=1/𝑝+1/𝑟, and let 𝑘∈{0,1,2,…,𝑠}. Assume that 𝐹(𝑣) is class of 𝐶^𝑠 and satisfies 𝜕^𝑙_𝑣𝐹(𝑣)≤𝐶_𝑙,𝛿|𝑣|^{𝜃+1−𝑙},|𝑣|𝜕≤𝛿,0≤𝑙≤𝑠,𝑙<𝜃+1,_𝑙𝑣𝐹(𝑣)≤𝐶_𝑙,𝛿,|𝑣|≤𝛿,𝑙≤𝑠,𝜃+1≤𝑙. (47) If 𝑣∈𝐿^𝑝⋂𝑊^𝑘,𝑞⋂𝐿^∞ and ‖𝑣‖_𝐿^∞≤𝛿, then for |𝛼|≤𝑘 one has ‖𝐹(𝑣)‖_𝑊^𝑘,𝑟≤𝐶_𝑘,𝛿‖𝑣‖_𝑊^𝑘,𝑞‖𝑣‖_𝐿^𝑝‖𝑣‖_𝐿^𝜃−1∞,𝜕^𝛼_𝑥𝐹(𝑣)_𝐿𝑟≤𝐶_𝑘,𝛿𝜕^𝛼_𝑥𝑣_𝐿𝑞‖𝑣‖_𝐿^𝑝‖𝑣‖_𝐿^𝜃−1∞. (48)

Lemma 6. Let 𝑠 and 𝜃 be positive integers, let 𝛿>0, 𝑝, 𝑞, 𝑟∈[1,∞] satisfy 1/𝑟=1/𝑝+1/𝑟, and let 𝑘∈{0,1,2,…,𝑠}. Let 𝐹(𝑣) be a function that satisfies the assumptions of Lemma 5. Moreover, assume that 𝜕^𝑠_𝑣𝐹𝑣₁−𝜕^𝑠_𝑣𝐹𝑣₂≤𝐶_𝛿𝑣₁+𝑣₂^{max{𝜃−𝑠,𝜃}}𝑣₁−𝑣₂,𝑣₁≤𝛿,𝑣₂≤𝛿. (49) If 𝑣₁, 𝑣₂∈𝐿^𝑝⋂𝑊^𝑘,𝑞⋂𝐿^∞ and ‖𝑣₁‖_𝐿∞≤𝛿, ‖𝑣₂‖_𝐿∞≤𝛿, then for |𝛼|≤𝑘, one has 𝜕^𝛼_𝑥𝐹𝑣₁−𝐹𝑣_2𝐿𝑟≤𝐶_𝑘,𝛿𝜕^𝛼_𝑥𝑣_1𝐿𝑞+𝜕^𝛼_𝑥𝑣_2𝐿𝑞𝑣₁−𝑣_2𝐿𝑝+𝑣_1𝐿𝑝+𝑣_2𝐿𝑝𝜕^𝛼_𝑥𝑣₁−𝑣_2𝐿𝑞×𝑣_1𝐿∞+𝑣_2𝐿∞^𝜃−1. (50)
Based on the estimates (42)–(44) of solutions to the linear problem (4), (2), one defines the following solution space: 𝑋=𝑢∈𝐶[0,∞);𝐻^𝑠+4ℝ^𝑛𝐶¹[0,∞);𝐻^𝑠ℝ^𝑛∶‖𝑢‖_𝑋<∞, (51) where ‖𝑢‖_𝑋=sup_{𝑡≥0𝑘≤𝑠+4}(1+𝑡)^{𝑛/8+𝑘/4}𝜕^𝑘_𝑥𝑢(𝑡)_𝐿2+_𝑙≤𝑠(1+𝑡)^{𝑛/8+𝑙/4+1/2}𝜕^𝑙_𝑥𝑢_𝑡(𝑡)_𝐿2. (52) For 𝑅>0, one defines 𝑋_𝑅=𝑢∈𝑋∶‖𝑢‖_𝑋≤𝑅, (53) where 𝑅 depends on the initial value, which is chosen in the proof of main result.
For ℎ≤𝑠−[𝑛/2]+3, using Gagliardo-Nirenberg inequality, one obtains 𝜕^ℎ_𝑥𝑢(𝑡)_𝐿∞≤𝐶(1+𝑡)^{−(𝑛/4+ℎ/4)}‖𝑢‖_𝑋. (54)

Theorem 7. Assume that 𝑢₀∈𝐻^𝑠+4(ℝ^𝑛)⋂𝐿¹(ℝ^𝑛), 𝑢₁∈𝐻^𝑠(ℝ^𝑛)⋂𝐻̇₁⁻²(ℝ^𝑛)𝑛𝑠≥2+1, and integer 𝜃≥1. 𝑓(𝑢) satisfies the assumptions of Lemmas 5 and 6. Put 𝐸₀=𝑢_0𝐿1+𝑢_1̇𝐻1−2+𝑢_0𝐻𝑠+4+𝑢_1𝐻𝑠. (55) If 𝐸₀ is suitably small, the Cauchy problem (1)-(2) has a unique global classical solution 𝑢(𝑥,𝑡) satisfying 𝑢∈𝐶[0,∞);𝐻^𝑠+4ℝ^𝑛,𝑢_𝑡∈𝐶[0,∞);𝐻^𝑠ℝ^𝑛. (56) Moreover, the solution satisfies the decay estimate 𝜕^𝑘_𝑥𝑢(𝑡)_𝐿2≤𝐶𝐸₀(1+𝑡)^{−(𝑛/8+𝑘/4)},𝜕^𝑙_𝑥𝑢_𝑡(𝑡)_𝐿2≤𝐶𝐸₀(1+𝑡)^{−(𝑛/8+𝑙/4+1/2)} (57) for 𝑘≤𝑠+4 and 𝑙≤𝑠.

Proof. Define the mapping 𝕋(𝑢)=𝐺(𝑡)∗𝑢₁+𝐻(𝑡)∗𝑢₀+^𝑡₀𝐺(𝑡−𝜏)∗Δ𝑓(Δ𝑢(𝜏))𝑑𝜏. (58) Using (34)-(35), (38), Lemma 5, and (54), for 𝑘≤𝑠+4, we obtain 𝜕^𝑘_𝑥𝕋(𝑢)_𝐿2≤𝐶𝜕^𝑘_𝑥𝐺(𝑡)∗𝑢_1𝐿2+𝐶𝜕^𝑘_𝑥𝐻(𝑡)∗𝑢_0𝐿2+𝐶^𝑡₀𝜕^𝑘_𝑥𝐺(𝑡−𝜏)∗Δ𝑓(Δ𝑢(𝜏))_𝐿2()𝑑𝜏≤𝐶1+𝑡^{−(𝑛/8+𝑘/4)}𝑢_1̇𝐻1−2+𝐶𝑒^−𝑐𝑡𝜕^(𝑘−4)+_𝑥𝑢_1𝐿2+𝐶(1+𝑡)^{−(𝑛/8+𝑘/4)}𝑢_0𝐿1+𝐶𝑒^−𝑐𝑡𝜕^𝑘_𝑥𝑢_0𝐿2+𝐶₀^𝑡/2(1+𝑡−𝜏)^{−(𝑛/8+𝑘/4)}𝑓(Δ𝑢)_𝐿1𝑑𝜏+𝐶^𝑡_𝑡/2(1+𝑡−𝜏)^−𝑛/8𝜕^𝑘_𝑥𝑓(Δ𝑢)_𝐿1𝑑𝜏+𝐶^𝑡₀𝑒^{−𝑐(𝑡−𝜏)}𝜕^(𝑘−2)+_𝑥𝑓(Δ𝑢)_𝐿2()𝑑𝜏≤𝐶1+𝑡^{−(𝑛/8+𝑘/4)}𝑢_0𝐿1+𝑢_1̇𝐻1−2+𝐶𝑒^−𝑐𝑡𝑢_0𝐻𝑠+4+𝑢_1𝐻𝑠+𝐶₀^𝑡/2(1+𝑡−𝜏)^{−(𝑛/8+𝑘/4)}‖Δ𝑢‖²_𝐿2‖Δ𝑢‖_𝐿^𝜃−1∞𝑑𝜏+𝐶^𝑡_𝑡/2(1+𝑡−𝜏)^{−𝑛/8−1/2}‖Δ𝑢‖_𝐿²×𝜕^(𝑘−2)+_𝑥Δ𝑢_𝐿2‖Δ𝑢‖_𝐿^𝜃−1∞𝑑𝜏+𝐶^𝑡₀𝑒^{−𝑐(𝑡−𝜏)}𝜕^(𝑘−2)+_𝑥Δ𝑢_𝐿2‖Δ𝑢‖^𝜃_𝐿^∞()𝑑𝜏≤𝐶1+𝑡^{−(𝑛/8+𝑘/4)}𝑢_0𝐿1+𝑢_1̇𝐻1−2+𝐶𝑒^−𝑐𝑡𝑢_0𝐻𝑠+4+𝑢_1𝐻𝑠+𝐶𝑅^𝜃+1₀^𝑡/2(1+𝑡−𝜏)^{−(𝑛/8+𝑘/4)}(1+𝜏)^{−(𝑛/4+1)}×(1+𝜏)^{−(𝑛/4+1/2)(𝜃−1)}𝑑𝜏+𝐶𝑅^𝜃+1𝑡_𝑡/2(1+𝑡−𝜏)^{−𝑛/8−1/2}(1+𝜏)^{−(𝑛/4+𝑘/4)}×(1+𝜏)^{−(𝑛/4+1/2)(𝜃−1)}𝑑𝜏+𝐶𝑅^𝜃+1𝑡₀𝑒^{−𝑐(𝑡−𝜏)}(1+𝜏)^{−(𝑛/8+((𝑘−2)++2)/4)}×(1+𝜏)^{−(𝑛/4+1/2)𝜃}()𝑑𝜏≤𝐶1+𝑡^{−(𝑛/8+𝑘/4)}×𝑢_0𝐿1+𝑢_1̇𝐻1−2+𝑢_0𝐻𝑠+4+𝑢_1𝐻𝑠+𝑅^𝜃+1. (59) Thus (1+𝑡)^{𝑛/8+𝑘/4}𝜕^𝑘_𝑥𝕋(𝑢)_𝐿2≤𝐶𝐸₀+𝐶𝑅^𝜃+1. (60) It follows from (58) that 𝕋(𝑢)_𝑡=𝐺_𝑡(𝑡)∗𝑢₁+𝐻_𝑡(𝑡)∗𝑢₀+^𝑡₀𝐺_𝑡(𝑡−𝜏)∗Δ𝑓(Δ𝑢(𝜏))𝑑𝜏. (61) Using (36)-(37), (39), Lemma 5, and (54), for 𝑙≤𝑠, we have 𝜕^𝑙_𝑥𝕋(𝑢)_𝑡𝐿2≤𝐶𝜕^𝑙_𝑥𝐺_𝑡(𝑡)∗𝑢_1𝐿2+𝐶𝜕^𝑙_𝑥𝐻_𝑡(𝑡)∗𝑢_0𝐿2+𝐶^𝑡₀𝜕^𝑙_𝑥𝐺_𝑡(𝑡−𝜏)∗Δ𝑓(Δ𝑢(𝜏))_𝐿2()𝑑𝜏≤𝐶1+𝑡^{−(𝑛/8+𝑙/4+1/2)}𝑢_1̇𝐻1−2+𝐶𝑒^−𝑐𝑡𝜕^𝑙_𝑥𝑢_1𝐿2+𝐶(1+𝑡)^{−(𝑛/8+𝑙/4+1/2)}𝑢_0𝐿1+𝐶𝑒^−𝑐𝑡𝜕^𝑙+4𝑢_0𝐿2+𝐶₀^𝑡/2(1+𝑡−𝜏)^{−(𝑛/8+𝑙/4+1/2)}𝑓(Δ𝑢)_𝐿1𝑑𝜏+𝐶^𝑡_𝑡/2(1+𝑡−𝜏)^{−(𝑛/8+1/2)}𝜕^𝑙_𝑥𝑓(Δ𝑢)_𝐿1𝑑𝜏+𝐶^𝑡₀𝑒^{−𝑐(𝑡−𝜏)}𝜕_𝑥^𝑙+2𝑓(Δ𝑢)_𝐿2()𝑑𝜏≤𝐶1+𝑡^{−(𝑛/8+𝑙/4+1/2)}×𝑢_0𝐿1+𝑢_1̇𝐻1−2+𝑢_0𝐻𝑠+4+𝑢_1𝐻𝑠+𝐶₀^𝑡/2(1+𝑡−𝜏)^{−(𝑛/8+𝑙/4+1/2)}‖Δ𝑢‖²_𝐿2‖Δ𝑢‖_𝐿^𝜃−1∞𝑑𝜏+𝐶^𝑡_𝑡/2(1+𝑡−𝜏)^{−(𝑛/8+1/2)}𝜕^𝑙_𝑥Δ𝑢²_𝐿2‖Δ𝑢‖_𝐿^𝜃−1∞𝑑𝜏+𝐶^𝑡₀𝑒^{−𝑐(𝑡−𝜏)}𝜕_𝑥^𝑙+4𝑢_𝐿2‖Δ𝑢‖^𝜃_𝐿^∞()𝑑𝜏≤𝐶1+𝑡^{−(𝑛/8+𝑙/4+1/2)}×𝑢_0𝐿1+𝑢_1̇𝐻1−2+𝑢_0𝐻𝑠+4+𝑢_1𝐻𝑠+𝐶𝑅^𝜃+1₀^𝑡/2(1+𝑡−𝜏)^{−(𝑛/8+𝑙/4+1/2)}(1+𝜏)^{−(𝑛/4+1)}×(1+𝜏)^{−(𝑛/4+1/2)(𝜃−1)}𝑑𝜏+𝐶𝑅^𝜃+1𝑡_𝑡/2(1+𝑡−𝜏)^{−(𝑛/8+1/2)}(1+𝜏)^{−(𝑛/4+𝑙/2+1)}×(1+𝜏)^{−(𝑛/4+1/2)(𝜃−1)}𝑑𝜏+𝐶𝑅^𝜃+1𝑡₀𝑒^{−𝑐(𝑡−𝜏)}(1+𝜏)^{−(𝑛/8+(𝑙+4)/4)}×(1+𝜏)^{−(𝑛/4+1/2)𝜃}()𝑑𝜏≤𝐶1+𝑡^{−(𝑛/8+𝑙/4+1/2)}×𝑢_0𝐿1+𝑢_1̇𝐻1−2+𝑢_0𝐻𝑠+4+𝑢_1𝐻𝑠+𝑅^𝜃+1. (62) Thus (1+𝑡)^{𝑛/8+𝑙/4+1/2}𝜕^𝑙_𝑥𝕋(𝑢)_𝑡𝐿2≤𝐶𝐸₀+𝐶𝑅^𝜃+1. (63) Combining (60), (63) and taking 𝑅=2𝐶𝐸₀ and 𝐸₀ suitably small yield ‖𝕋(𝑢)‖_𝑋≤2𝐶𝐸₀. (64) For 𝑢, 𝑢∈𝑋_𝑅, by using (58), we have 𝕋(𝑢)−𝕋(𝑢)=^𝑡₀𝐺(𝑡−𝜏)∗Δ𝑓(Δ𝑢)−𝑓(Δ𝑢)𝑑𝜏. (65) Exploiting (65), (38) Lemma 6, and (54), for 𝑘≤𝑠+4, we obtain 𝜕^𝑘_𝑥(𝕋(𝑢)−𝕋(𝑢))_𝐿2≤^𝑡₀𝜕^𝑘_𝑥𝐺(𝑡−𝜏)∗Δ𝑓(Δ𝑢)−𝑓(Δ𝑢)_𝐿2𝑑𝜏≤𝐶₀^𝑡/2(1+𝑡−𝜏)^{−(𝑛/8+𝑘/4)}𝑓(Δ𝑢)−𝑓(Δ𝑢)_𝐿1𝑑𝜏+𝐶^𝑡_𝑡/2(1+𝑡−𝜏)^−𝑛/8𝜕^𝑘_𝑥𝑓(Δ𝑢)−𝑓(Δ𝑢)_𝐿1𝑑𝜏+𝐶^𝑡₀𝑒^{−𝑐(𝑡−𝜏)}𝜕^(𝑘−2)+_𝑥𝑓(Δ𝑢)−𝑓(Δ𝑢)_𝐿2𝑑𝜏≤𝐶₀^𝑡/2(1+𝑡−𝜏)^{−(𝑛/8+𝑘/4)}‖Δ𝑢‖_𝐿²+‖Δ𝑢‖_𝐿²×‖Δ(𝑢−𝑢)‖_𝐿²‖Δ𝑢‖_𝐿^∞+‖Δ𝑢‖_𝐿^∞𝜃−1𝑑𝜏+𝐶^𝑡_𝑡/2(1+𝑡−𝜏)^{−𝑛/8−1/2}×𝜕^(𝑘−2)+_𝑥Δ𝑢_𝐿2+𝜕^(𝑘−2)+_𝑥Δ𝑢_𝐿2‖Δ(𝑢−𝑢)‖_𝐿²+‖Δ𝑢‖_𝐿²+‖Δ𝑢‖_𝐿²𝜕^(𝑘−2)+_𝑥Δ(𝑢−𝑢)_𝐿2×‖Δ𝑢‖_𝐿^∞+‖Δ𝑢‖_𝐿^∞𝜃−1𝑑𝜏+𝐶^𝑡₀𝑒^{−𝑐(𝑡−𝜏)}×𝜕^(𝑘−2)+_𝑥Δ𝑢_𝐿2+𝜕^(𝑘−2)+_𝑥Δ𝑢_𝐿2×‖Δ(𝑢−𝑢)‖_𝐿^∞+‖Δ𝑢‖_𝐿^∞+‖Δ𝑢‖_𝐿^∞×𝜕^(𝑘−2)+_𝑥Δ(𝑢−𝑢)_𝐿2×‖Δ𝑢‖_𝐿^∞+‖Δ𝑢‖_𝐿^∞𝜃−1𝑑𝜏≤𝐶𝑅^𝜃‖𝑢−𝑢‖_𝑋0^𝑡/2(1+𝑡−𝜏)^{−(𝑛/8+𝑘/4)}×(1+𝜏)^{−(𝑛/4+1+(𝑛/4+1/2)(𝜃−1))}𝑑𝜏+𝐶𝑅^𝜃‖𝑢−𝑢‖_𝑋^𝑡_𝑡/2(1+𝑡−𝜏)^{−𝑛/8−1/2}×(1+𝜏)^{−(𝑛/4+𝑘/4+1/2+(𝑛/4+1/2)(𝜃−1))}𝑑𝜏+𝐶𝑅^𝜃‖𝑢−𝑢‖_𝑋^𝑡₀𝑒^{−𝑐(𝑡−𝜏)}×(1+𝜏)^{−(𝑛/8+((𝑘−2)++2)/4+(𝑛/4+1/2)𝜃)}𝑑𝜏≤𝐶𝑅^𝜃(1+𝑡)^{−(𝑛/8+𝑘/4)}‖𝑢−𝑢‖_𝑋, (66) which implies (1+𝑡)^{𝑛/8+𝑘/4}𝜕^𝑘_𝑥(𝕋(𝑢)−𝕋(𝑢))_𝐿2≤𝐶𝑅^𝜃‖𝑢−𝑢‖_𝑋. (67) Similarly for 𝑙≤𝑠, from (61), (39), and (54), we have 𝜕^𝑙_𝑥(𝕋(𝑢)−𝕋(𝑢))_𝑡𝐿2≤^𝑡₀𝜕^𝑙_𝑥𝐺_𝑡(𝑡−𝜏)∗Δ𝑓(Δ𝑢)−𝑓(Δ𝑢)_𝐿2𝑑𝜏≤𝐶₀^𝑡/2(1+𝑡−𝜏)^{−(𝑛/8+𝑙/4+1/2)}𝑓(Δ𝑢)−𝑓(Δ𝑢)_𝐿1𝑑𝜏+𝐶^𝑡_𝑡/2(1+𝑡−𝜏)^{−(𝑛/8+1/2)}𝜕^𝑙_𝑥𝑓(Δ𝑢)−𝑓(Δ𝑢)_𝐿1𝑑𝜏+𝐶^𝑡₀𝑒^{−𝑐(𝑡−𝜏)}𝜕_𝑥^𝑙+2𝑓(Δ𝑢)−𝑓(Δ𝑢)_𝐿2𝑑𝜏≤𝐶𝑅^𝜃(1+𝑡)^{−(𝑛/8+𝑙/4+1/2)}‖𝑢−𝑢‖_𝑋, (68) which implies (1+𝑡)^{𝑛/8+𝑙/4+1/2}𝜕^𝑙_𝑥(𝕋(𝑢)−𝕋(𝑢))_𝑡𝐿2≤𝐶𝑅^𝜃‖𝑢−𝑢‖_𝑋. (69) Noting 𝑅=2𝐶𝐸₀, by using (67), (69) and taking 𝐸₀ suitably small, yields ‖𝕋(𝑢)−𝕋(𝑢)‖_𝑋≤12‖𝑢−𝑢‖_𝑋. (70)
From (64) and (70), we know that 𝕋 is strictly contracting mapping. Consequently, we conclude that there exists a fixed point 𝑢∈𝑋_𝑅 of the mapping 𝕋, which is a classical solution to (1), (2). We have completed the proof of Theorem 7.

Acknowledgments

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