Exact Null Controllability of KdV-Burgers Equation with Memory Effect Systems

Abstract

This paper is concerned with exact null controllability analysis of nonlinear KdV-Burgers equation with memory. The proposed approach relies upon regression tool to prove controllability property of linearized KdV-Burgers equation via Carleman estimates. The control is distributed along with subdomain 𝜔⊂Ω and the external control acts on the key role of observability inequality with memory. This description finally showed the exact null controllability guaranteeing the stability.

1. Introduction

In recent years there has been rapidly increasing interest in mathematical studies of dynamical and statistical property of nonlinear fields described by the Burgers equation (see [1–4]) and it has been motivated by several developments. As Burgers [5] noticed, the Burgers equation is a convenient analytical model for the physical turbulence, which is simultaneously taken into an account of two competing mechanisms: to determine properties of the strong hydrodynamic turbulence: the interior nonlinearity and viscosity.

Moreover, the quantitative description in many physical processes leads to the Burgers equation. One example here is an intense acoustical noise, such as the jet noise [6], where knowledge of dynamical and statistical properties of the Burgers turbulence can be directly applied to an analysis of nonlinear distortions. Another phenomenon is adequately for the nonlinear evolution of gravitational instability and the related characteristic of large-scale cellular structures (see [1, 7]).

Many problems have been modelled on nonlinear Burgers equations. There has been an enormous on-going research and it is directly investigating some nonlinear effects. For example, the simpler model equation with higher dimension, which encapsulates to essential features of the problem. But it is impossible to solve it directly in a higher dimension. After few years, these issues were carried out by Fernández-Cara et al. [8] and it has been discussed via Korteweg-de Vries (KdV). The KdV equation is a prototype of such a model also describing the competition between nonlinear and disperse effects in water waves.

The KdV equations are not directly related to the disperse waves, the density fields in Burgers turbulence, and it is rather difficult to do in higher dimensions. Although the KdV equation of analytical model is not a convenient and also nonefficient model of the strong turbulence, it holds at least one serious drawback from the viewpoint of physical applications (see [6]). Namely, it does not take into account pressure forces that could lead to smoothing of singularity, which appear in density fields driven by Burgerian velocity.

In this context we will consider the KdV-Burgers equations with memory, which is described by the density fields, pressure forces as well as the viscosity and its dispersion, and so forth. In various fields of physics and engineering, problems have been modelled by partial differential equation with memory (see [9]). It is essential to take an account of the effect of past history.

We consider a typical form of KdV-Burgers equation with memory; it is essential and suitable for the above physical situation: 𝑦_𝑡−𝑦_𝑥𝑥+𝑦𝑦_𝑥+𝑦_𝑥𝑥𝑥+^𝑡₀𝑘(𝑡,𝜏)𝑦_𝑥𝑥𝑑𝜏=𝜒_𝜔𝑢()(),()∈()𝑦()𝑡,𝑥+𝑓𝑡,𝑥𝑡,𝑥0,𝑇×Ω,0,𝑥=𝑦₀(𝑥)𝑦()()∈(),𝑥∈Ω,𝑡,𝑥=0,𝑡,𝑥0,𝑇×𝜕Ω, (1.1) where Ω⊂𝑅^𝑛 is a bounded domain with smooth boundary 𝜕Ω. The null controllability of linear parabolic equation without memory kernels has been extensively studied by several authors (see [8, 10–12], and references therein). Barbu and Iannelli [9] discussed the approximate controllability for the similar type of equation with memory. Rosier [13] studied the exact boundary controllability of linear KdV equation with the half-line and Russell and Zhang[14] also discussed the exact controllability and stabilizability of KdV equation. Sakthivel [15] has been proved the asymptotic stability of KdV-Burgers via Lyapunov function technique by using 𝐿₂ and 𝐿_∞ norms with domain of [0,1]. But the domain of attraction always containing [0,1] is impossible; therefore, the problems of KdV-Burgers equations have not been fully investigated, and it is therefore still a challenging problem.

Some nonlinear control systems are modelled by partial differential equations; it will be a strong control. Unfortunately in some cases, even if the linearized control system around the equilibrium is not controllable. But the linear control system around the memory kernel is always controllable. This method has been introduced in Temam [16], Kofman et al. [7], and Coron [17] where they have discussed about the controllability depending on the relationship between the pressure forces and the magnitude of initial velocity in gas models. The distributed control is described by the behavior of water waves in a shallow channel, compressible gas and strong hydrodynamic turbulence, and so forth. It will be indicated as in (1.2) that appears as in control function.

We consider the linearized control KDV-Burgers equation with memory effects of dirichlet boundary conditions, which helps to develop (1.1) (as based on the existing result Fursikov and Imanuvilov [12]): 𝑦_𝑡(𝑡,𝑥)−𝑦_𝑥𝑥(𝑡,𝑥)+𝑎(𝑡,𝑥)𝑦(𝑡,𝑥)_𝑥+𝑏(𝑡,𝑥)𝑦(𝑡,𝑥)+𝑦_𝑥𝑥𝑥(𝑡,𝑥)+^𝑡₀𝑘(𝑡,𝜏)𝑦_𝑥𝑥(𝜏,𝑥)𝑑𝜏=𝜒_𝜔𝑢()(),()∈()𝑦()𝑡,𝑥+𝑓𝑡,𝑥𝑡,𝑥0,𝑇×Ω,0,𝑥=𝑦₀(𝑥)𝑦()()∈(),𝑥∈Ω𝑡,𝑥=0,𝑡,𝑥0,𝑇×𝜕Ω, (1.2) where 𝜕³𝑦/𝜕𝑥³=𝑦_𝑥𝑥𝑥,𝑦(𝑡,𝑥) is a solitary wave of dispersion at the point 𝑥 and time 𝑡, 𝑦₀(𝑥) is an initial temperature distribution, the integral kernel 𝑘(𝑡,𝜏) is called conservation of mass or volume and has support in (𝑡₁,𝑡₂), where 0<𝑡₁<𝑡₂<𝑇, 𝑓(𝑡,𝑥) is a forcing term such as pressure force, 𝜒_𝜔 is a characteristic function of the subset 𝜔⊂Ω, and 𝑢(𝑡,𝑥) is a control over an arbitrary sub domain 𝜔 of the domain Ω.

The paper is organized as follows. Section 2 gives some basic assumptions and formation of the problem. Section 3 gives the proof of the Carleman estimate and observability result. Section 4 gives exact null controllability result as based on a unique continuation result of adjoined problems and by using the observability inequality. A conclusion will be given in Section 5.

Notation. We describe some function spaces which will be useful to formulate our results. For each positive integer 𝑚 and 𝑝>1 or 𝑝=∞, denote as usual by 𝑊^𝑚,𝑝(Ω) the sobolev space of functions in 𝐿^𝑝(Ω) whose weak derivatives are of order less than or equal to 𝑚. When 𝑝=2 instead of 𝑊^𝑚,𝑝(Ω), we will write 𝐻^𝑚(Ω). Besides, we need the space 𝐿²(0,𝑇∶𝐻¹(Ω)) of all equivalence classes of square integrable functions from (0,𝑇) to 𝐻¹(Ω). The spaces 𝐿²(0,𝑇∶𝐿²(Ω)) and 𝐿^∞(0,𝑇∶𝐿²(Ω)) are analogously defined. Moreover, we set (see [18]) (i)Ω={(𝑡,𝑥)∶0<𝑡<𝑇,0<𝑥<∞}; (ii)𝜒_𝜔∶=1for𝑥∈𝜔,0for𝑥∈Ω/𝜔;(iii)𝑄∶=(0,𝑇)×Ω,𝑄_𝜔∶=(0,𝑇)×𝜔,andΣ∶=(0,𝑇)×𝜕Ω; (iv)𝐻^𝑚(Ω):= the Sobolev spaces of functions in 𝐿²(Ω) whose weak derivatives are of order less than or equal to 𝑚, where 𝑚 is a positive integer; (v)𝐿²(0,𝑇;𝐻¹(Ω)):= the space of all equivalence classes of square integrable functions from (0,𝑇) to 𝐻¹(Ω);(vi)𝑊^𝑚,𝑝(Ω)={𝑦(𝑥)∶‖𝑦‖_{𝑊𝑚,𝑝}=(∑_{|𝛼|≤𝑚}∫_Ω|𝐷^𝛼𝑦|^𝑝𝑑𝑥)^1/𝑝<∞}.

2. Assumptions and Main Results

A linearized control system (1.2) is a weak solution model of the control problem (1.1) (it is supporting to prove the stabilization), then system (1.2) can be written as 𝑦_𝑡(𝑡,𝑥)−𝑦_𝑥𝑥(𝑡,𝑥)+𝑎(𝑡,𝑥)𝑦_𝑥(𝑡,𝑥)+𝑏(𝑡,𝑥)𝑦(𝑡,𝑥)+𝑦_𝑥𝑥𝑥(𝑡,𝑥)+^𝑡₀𝑘(𝑡,𝜏)𝑦_𝑥𝑥(𝜏,𝑥)𝑑𝜏=𝜒_𝜔𝑢()(),()∈()𝑦()𝑡,𝑥+𝑓𝑡,𝑥𝑡,𝑥0,𝑇×Ω,0,𝑥=𝑦₀(𝑥)𝑦()()∈(),𝑥∈Ω,𝑡,𝑥=0,𝑡,𝑥0,𝑇×𝜕Ω, (2.1) where 𝑎∈𝑊^1,∞(0,𝑇∶𝐿²(Ω)),𝑏∈𝐿^∞(𝑄). The kernel 𝑘(𝑡,⋅) is smooth and has support in (𝑡₀,𝑡₁) where 0<𝑡₀<𝑡₁<𝑇. Moreover, there are plenty of works related to exact and approximate controllability properties of the parabolic system of type (1.2) without the memory effects (see [11]). In this paper we constructed a system with memory; finally by using Hölder’s inequality and changing the order of integration the memory term will be observed by 𝑦_𝑥𝑥∈𝜔⊂Ω.

The following lemma is a fundamental tool to proving controllability results.

Lemma 2.1. Let Ω be an open bounded and connected subset of the boundary 𝜕Ω in class 𝐶², 𝜔 an arbitrary subsets of Ω, and 𝑓∈𝐿²(𝑄) such that 𝑓(𝑥,𝑡)≤𝑓₁(𝑥,𝑡)𝑒^{𝑠𝜑(𝑥,𝑡)}𝜑_𝑥(𝑥,𝑡)𝜑_𝑥𝑥(𝑥,𝑡),a.e.(𝑥,𝑡)∈𝑄. (2.2) For each 𝑦₀∈𝐿²(Ω),and𝑢∈𝐿²(𝑄) such that ‖𝑢‖_𝐿²_(𝑄)≤𝐶𝑦₀₂+𝑓_1𝐿2(𝑄), (2.3) and the space 𝑦^𝑢∈𝐶¹[0,𝑇];𝐻¹([0,𝑇]∶𝐿²(Ω))∩𝐿²(0,𝑇;𝐻¹(Ω)) is a solution of (1.2).

Proof. The proof is similar to Theorem 1.3.1 from Barbu [10] and hence it will be omitted.

Theorem 2.2. Let Ω,𝜔 be as in Lemma 2.1. System (1.2) is exactly null controllable for each 𝑇>0, if there exists 𝑢∈𝐿²(𝑄) and 𝑦^𝑢∈𝐶([0,𝑇];𝐿²(Ω))∩𝐿²(0,𝑇;𝐻¹(Ω)) such that 𝑦^𝑢(𝑇,𝑥)=0a.e.𝑥∈Ω,‖𝑢‖_𝐿²_(𝑄)≤𝐶𝑦₀₂+𝑓₁²_𝐿2(𝑄). (2.4)

(The proof of the theorem is given in Section 4).

3. Observability Results and Carleman Estimates

In this section we will derive the observability result via the Carleman estimate of adjoint system (2.1).

To prove the Carleman estimate, the weight functions are necessary. As based on Fernández-Cara et al. [8] and Rosier [13] the weight functions 𝜑 and 𝜓 are assumed as 𝜑()=𝜓(𝑥)𝑥,𝑡𝛽(𝑡),𝜓(𝑥)>0,∀𝑥∈Ω,𝜓(𝑥)=0,∀𝑥∈𝜕Ω,𝜓_𝑥(𝑥𝜓)>0,_𝑥(𝑥)<0,𝜓_𝑥𝑥(𝑥)<0,𝜓_𝑥𝑥(𝑥)⋅𝜓_𝑥(𝑥)𝜓>0,_𝑥(𝑥)⋅𝜓_𝑥𝑥𝑥(𝑥)>0,𝑥∈Ω⧵𝜔, (3.1) where 𝛽(𝑡)=𝑡(𝑇−𝑡).

Suppose that 𝑤 is an adjoint state variable of system (2.1), then it has some solutions. Therefore, 𝑤_𝑡(𝑡,𝑥)+𝑤_𝑥𝑥(𝑡,𝑥)+𝑐₁𝑤_𝑥(𝑡,𝑥)−𝑐₂𝑤(𝑡,𝑥)+𝑤_𝑥𝑥𝑥(𝑡,𝑥)−^𝑇_𝑡𝑘(𝜏,𝑡)𝑤_𝑥𝑥()(),()𝑤()𝜏,𝑥𝑑𝜏=𝑔𝑡,𝑥𝑡,𝑥∈Σ,𝑇,𝑥=𝑤_𝑇(𝑥)𝑤()()∈(),𝑥∈Ω,𝑡,𝑥=0,𝑡,𝑥0,𝑇×𝜕Ω, (3.2) where ‖𝑎(𝑡,𝑥)‖≤𝑐₁>0,‖𝑏(𝑡,𝑥)‖≤𝑐₂>0,𝑔∈𝐿²(𝑄) and 𝑤_𝑇∈𝐿²(Ω).

Now we will state the theorem to prove system (3.2) of solution 𝑤,whichwill be a solution for (2.1). The question is how the system (3.2) of solution 𝑤 will be a solution for (2.1). As based on Carleman estimate (assume the weight functions 𝜑,𝜓 as (3.1)), we can prove that 𝑤 will be a solution for (2.1) (see [12]).

Theorem 3.1 (Carleman estimate). The function 𝜓 defined as in (3.1), the kernel has support in (𝑡₁,𝑡₂) (where 0<𝑡₁<𝑡₂<𝑇), there exist positive constants 𝐶, and 𝑠≥𝑠₀ such that the following inequalities hold when 𝑤 a solution of (3.2): _𝑄𝑒^2𝑠𝜑𝑠𝜑⁻¹𝑤_𝑡²+𝑤_𝑥²+𝑤_𝑥𝑥²+𝑤_𝑥𝑥𝑥²+𝑠⁵𝜑⁴_𝑥𝜑_𝑥𝑥|𝑤|²+𝑠³𝜑²_𝑥𝜑_𝑥𝑥𝑤_𝑥²+𝑠𝜑𝑤_𝑥𝑥²𝑑𝑥𝑑𝑡≤𝐶𝑒^2𝑠𝜑_𝑄𝑔²𝑑𝑥𝑑𝑡+_𝑄𝜔𝑠⁴𝜑²_𝑥𝜑²_𝑥𝑥|𝑤|²𝑑𝑥𝑑𝑡. (3.3)

Proof. We set 𝑤=𝑒^−𝑠𝜑𝑞 (where 𝑠,𝜑 are positive parameters) and 𝑞 satisfies (3.2) (for our convenient 𝑐₁=𝑐₂=1), then 𝑞_𝑡+𝑠𝜑_𝑡𝑞+𝑠²𝜑²_𝑥𝑞−𝑠𝜑_𝑥𝑥𝑞−2𝑠𝜑_𝑥𝑞_𝑥+𝑞_𝑥𝑥−𝑠³𝜑³_𝑥𝑞+3𝑠²𝜑_𝑥𝜑_𝑥𝑥𝑞+3𝑠²𝜑²_𝑥𝑞_𝑥−3𝑠𝜑_𝑥𝑥𝑞_𝑥−3𝑠𝜑_𝑥𝑞_𝑥𝑥−𝑠𝜑_𝑥𝑥𝑥𝑞+𝑞_𝑥𝑥𝑥−𝑞−𝑠𝜑_𝑥𝑞+𝑞_𝑥−𝑒^𝑠𝜑𝑇_𝑡𝑘(𝜏,𝑡)𝑒^{−𝑠𝜑(𝜏)}𝑞(𝜏,𝑥)𝑑𝜏=𝑔𝑒^𝑠𝜑. (3.4) As based on Barbu [10] we can introduce the operators 𝐿₁𝑞=𝑞_𝑡−𝑠𝜑_𝑡𝑞−𝑠𝜑_𝑥𝑥𝑞+𝑠²𝜑²_𝑥𝑞−𝑠³𝜑³_𝑥𝑞+3𝑠²𝜑_𝑥𝜑_𝑥𝑥𝑞−𝑠𝜑_𝑥𝑥𝑥𝑞−𝑞−𝑠𝜑_𝑥𝐿𝑞,₂𝑞=𝑞_𝑥𝑥+𝑞_𝑥𝑥𝑥−2𝑠𝜑_𝑥𝑞_𝑥+3𝑠²𝜑²_𝑥𝑞_𝑥−3𝑠𝜑_𝑥𝑥𝑞_𝑥−3𝑠𝜑_𝑥𝑞_𝑥𝑥+𝑞_𝑥. (3.5)(3.6) It follows from (3.4), (3.5), and (3.6) that 𝐿₁𝑞+𝐿₂𝑞=ℎ_𝑠in𝑄, (3.7) where ℎ_𝑠=𝑔𝑒^𝑠𝜑+𝑒^𝑠𝜑𝑇_𝑡𝑘(𝜏,𝑡)𝑒^−𝑠𝜑𝑞_𝑥𝑥𝑑𝜏. (3.8) Taking 𝐿₂-norm of both sides of (3.7), we obtain ℎ_𝑠²_𝐿2(𝑄)=𝐿₁𝑞²_𝐿2(𝑄)+𝐿₂𝑞²_𝐿2(𝑄)+2𝐿₁𝑞,𝐿₂𝑞_𝐿2(𝑄). (3.9) Let us analyze the scalar product in (3.9) as 𝐿₁𝑞,𝐿₂𝑞_𝐿2=𝐸_𝑖𝑗,1≤𝑖≤9,1≤𝑗≤7, (3.10) where 𝐸_𝑖𝑗 is an integral product of 𝑖th term in 𝐿₁𝑞 and 𝑗th term in 𝐿₂𝑞.
Now we will simplify the estimate ⟨𝐿₁𝑞,𝐿₂𝑞⟩_𝐿2(𝑄) by using Green’s theorem with integration by parts: 𝐿₁𝑞,𝐿₂𝑞_𝐿2(𝑄)=_Σ𝑠𝜑_𝑥𝑥𝑥𝜕𝑞𝜕𝑣²3𝑑Σ+2_Σ𝑠²𝜑_𝑥𝜑_𝑥𝑥𝜕𝑞𝜕𝑣²1𝑑Σ+2_Σ𝑠³𝜑³_𝑥𝜕𝑞𝜕𝑣²−1𝑑Σ2_Σ𝑠²𝜑²_𝑥𝜕𝑞𝜕𝑣²1𝑑Σ+2_Σ𝑠𝜑_𝑥𝑥𝜕𝑞𝜕𝑣²1𝑑Σ+2_Σ𝑠𝜑_𝑡𝜕𝑞𝜕𝑣²𝑑Σ+_𝑄𝑠𝜑_𝑡𝑞²_𝑥−1𝑑𝑥𝑑𝑡2_𝑄𝑠𝜑_𝑡𝑥𝑥𝑞²1𝑑𝑥𝑑𝑡+2_𝑄𝑠𝜑_{𝑡𝑥𝑥𝑥}𝑞²𝑑𝑥𝑑𝑡−2_𝑄𝑠𝜑_𝑡𝑥𝑞²_𝑥𝑑𝑥𝑑𝑡+_𝑄𝑠𝜑_𝑡𝑞²_𝑥𝑑𝑥𝑑𝑡+_𝑄𝑠²𝜑_𝑡𝑥𝜑_𝑥𝑞²𝑑𝑥𝑑𝑡+_𝑄𝑠²𝜑_𝑡𝜑_𝑥𝑥𝑞²3𝑑𝑥𝑑𝑡+2_𝑄𝑠²𝜑_𝑡𝜑_𝑥𝑥𝑞²+3𝑑𝑥𝑑𝑡2_𝑄𝑠²𝜑_𝑡𝜑_𝑥𝑥𝑞²𝑑𝑥𝑑𝑡+3_𝑄𝑠³𝜑_𝑡𝜑_𝑥𝜑_𝑥𝑥𝑥𝑞²3𝑑𝑥𝑑𝑡+2_𝑄𝑠²𝜑_𝑡𝑥𝑞²𝑑𝑥𝑑𝑡+3_𝑄𝑠²𝜑_𝑡𝜑_𝑥𝑞²_𝑥3𝑑𝑥𝑑𝑡−2_𝑄𝑠²𝜑_𝑡𝑥𝑥𝜑_𝑥𝑞²𝑑𝑥𝑑𝑡+2_𝑄𝑠𝜑_𝑥𝑥𝑞²_𝑥𝑑𝑥𝑑𝑡−_𝑄𝑠²𝜑²_𝑥𝑥𝑞²3𝑑𝑥𝑑𝑡+2_𝑄𝑠³𝜑²_𝑥𝜑_𝑥𝑥𝑥𝑞²𝑑𝑥𝑑𝑡+3_𝑄𝑠³𝜑_𝑥𝜑²_𝑥𝑥𝑞²𝑑𝑥𝑑𝑡−3_𝑄𝑠²𝜑_𝑥𝜑_𝑥𝑥𝑞²_𝑥𝑑𝑥𝑑𝑡−2_𝑄𝑠²𝜑²_𝑥𝑞²_𝑥𝑑𝑥𝑑𝑡+_𝑄𝑠²𝜑²_𝑥𝑥𝑞²𝑑𝑥𝑑𝑡+2_𝑄𝑠²𝜑_𝑥𝜑_𝑥𝑥𝑞²−𝑑𝑥𝑑𝑡_𝑄𝑠²𝜑_𝑥𝑥𝜑_𝑥𝑥𝑥𝑞²𝑑𝑥𝑑𝑡−2_𝑄𝑠⁴𝜑³_𝑥𝜑_𝑥𝑥𝑞²𝑑𝑥𝑑𝑡+3_𝑄𝑠³𝜑³_𝑥𝑞²𝑑𝑥𝑑𝑡−9_𝑄𝑠³𝜑_𝑥𝜑²_𝑥𝑥𝑞²_𝑥3𝑑𝑥𝑑𝑡−2_𝑄𝑠³𝜑²_𝑥𝜑_𝑥𝑥𝑞²𝑑𝑥𝑑𝑡+2_𝑄𝑠³𝜑³_𝑥𝑞²_𝑥𝑑𝑥𝑑𝑡+7_𝑄𝑠³𝜑_𝑥𝜑_𝑥𝑥𝜑_𝑥𝑥𝑥𝑞²3𝑑𝑥𝑑𝑡+2_𝑄𝑠⁵𝜑⁴_𝑥𝜑_𝑥𝑥𝑞²𝑑𝑥𝑑𝑡+9_𝑄𝑠⁴𝜑²_𝑥𝜑²_𝑥𝑥𝑞²+3𝑑𝑥𝑑𝑡2_𝑄𝑠⁴𝜑³_𝑥𝜑_𝑥𝑥𝑥𝑞²𝑑𝑥𝑑𝑡−3_𝑄𝑠⁴𝜑⁴_𝑥𝑞²_𝑥𝑑𝑥𝑑𝑡+6_𝑄𝑠⁴𝜑³_𝑥𝜑_𝑥𝑥𝑥𝑞²𝑑𝑥𝑑𝑡+18_𝑄𝑠⁴𝜑²_𝑥𝜑²_𝑥𝑥𝑞²𝑑𝑥𝑑𝑡−6_𝑄𝑠²𝜑_𝑥𝜑_𝑥𝑥𝑞²_𝑥𝑑𝑥𝑑𝑡+3_𝑄𝑠²𝜑_𝑥𝑥𝜑²_𝑥𝑥𝑥𝑞²+3𝑑𝑥𝑑𝑡2_𝑄𝑠²𝜑_𝑥𝑥𝜑_𝑥𝑥𝑥𝑞²𝑑𝑥𝑑𝑡+6_𝑄𝑠²𝜑_𝑥𝑥𝜑_𝑥𝑥𝑥𝑞²_𝑥𝑑𝑥𝑑𝑡+3_𝑄𝑠²𝜑_𝑥𝑥𝜑_𝑥𝑥𝑥𝑞²_𝑥𝑑𝑥𝑑𝑡+4_𝑄𝑠²𝜑²_𝑥𝑥𝑞²𝑑𝑥𝑑𝑡+6_𝑄𝑠³𝜑_𝑥𝜑²_𝑥𝑥𝑞²9𝑑𝑥𝑑𝑡−2_𝑄𝑠⁴𝜑³_𝑥𝜑_𝑥𝑥𝑥𝑞²−𝑑𝑥𝑑𝑡272_𝑄𝑠⁴𝜑²_𝑥𝜑_𝑥𝑥𝑞²9𝑑𝑥𝑑𝑡+2_𝑄𝑠³𝜑³_𝑥𝑥𝑞²𝑑𝑥𝑑𝑡+9_𝑄𝑠³𝜑²_𝑥𝜑_𝑥𝑥𝑞²_𝑥−𝑑𝑥𝑑𝑡152_𝑄𝑠³𝜑³_𝑥𝑥𝑞²𝑑𝑥𝑑𝑡+_𝑄𝑠𝜑_𝑥𝑥𝑥𝑞²_𝑥𝑑𝑥𝑑𝑡+2_𝑄𝑠𝜑_𝑥𝑥𝑥𝑞²_𝑥𝑑𝑥𝑑𝑡−6_𝑄𝑠³𝜑²_𝑥𝜑_𝑥𝑥𝑞²_𝑥𝑑𝑥𝑑𝑡−_𝑄𝑠²𝜑_𝑥𝑥𝜑_𝑥𝑥𝑥𝑞²𝑑𝑥𝑑𝑡−3_𝑄𝑠²𝜑_𝑥𝜑_𝑥𝑥𝑥𝑞²_𝑥𝑑𝑥𝑑𝑡. (3.11) Since 𝑞=𝜑=0on𝜕Ω, some manipulations are dominated by the same parameters and also observed by powers of 𝑠. Finally we obtain ⁹⁷_𝑖=1𝑗=1𝐸_𝑖𝑗=𝐷₁+𝐷₂+32_𝑄𝑠⁵𝜑⁴_𝑥𝜑_𝑥𝑥𝑞²𝑑𝑥𝑑𝑡+3_𝑄𝑠³𝜑²_𝑥𝜑_𝑥𝑥𝑞²_𝑥𝑑𝑥𝑑𝑡, (3.12) where 𝐷₁,𝐷₂ are boundary terms: 𝐷₁=_Σ𝑠𝜑_𝑥𝑥𝑥𝜕𝑞𝜕𝑣²3𝑑Σ+2_Σ𝑠²𝜑_𝑥𝜑_𝑥𝑥𝜕𝑞𝜕𝑣²1𝑑Σ+2_Σ𝑠³𝜑³_𝑥𝜕𝑞𝜕𝑣²−1𝑑Σ2_Σ𝑠²𝜑²_𝑥𝜕𝑞𝜕𝑣²1𝑑Σ+2_Σ𝑠𝜑_𝑥𝑥𝜕𝑞𝜕𝑣²1𝑑Σ+2_Σ𝑠𝜑_𝑡𝜕𝑞𝜕𝑣²𝐷𝑑Σ,₂=_𝑄𝑠𝜑_𝑡𝑞²_𝑥1𝑑𝑥𝑑𝑡−2_𝑄𝑠𝜑_𝑡𝑥𝑥𝑞²1𝑑𝑥𝑑𝑡+2_𝑄𝑠𝜑_{𝑡𝑥𝑥𝑥}𝑞²𝑑𝑥𝑑𝑡−2_𝑄𝑠𝜑_𝑡𝑥𝑞²_𝑥𝑑𝑥𝑑𝑡+_𝑄𝑠𝜑_𝑡𝑞²_𝑥𝑑𝑥𝑑𝑡+_𝑄𝑠²𝜑_𝑡𝑥𝜑_𝑥𝑞²𝑑𝑥𝑑𝑡+_𝑄𝑠²𝜑_𝑡𝜑_𝑥𝑥𝑞²+3𝑑𝑥𝑑𝑡2_𝑄𝑠²𝜑_𝑡𝜑_𝑥𝑥𝑞²3𝑑𝑥𝑑𝑡+2_𝑄𝑠²𝜑_𝑡𝜑_𝑥𝑥𝑞²𝑑𝑥𝑑𝑡+3_𝑄𝑠³𝜑_𝑡𝜑_𝑥𝜑_𝑥𝑥𝑥𝑞²+3𝑑𝑥𝑑𝑡2_𝑄𝑠²𝜑_𝑡𝑥𝑞²𝑑𝑥𝑑𝑡+3_𝑄𝑠²𝜑_𝑡𝜑_𝑥𝑞²_𝑥3𝑑𝑥𝑑𝑡−2_𝑄𝑠²𝜑_𝑡𝑥𝑥𝜑_𝑥𝑞²𝑑𝑥𝑑𝑡+2_𝑄𝑠²𝜑_𝑥𝜑_𝑥𝑥𝑞²𝑑𝑥𝑑𝑡−_𝑄𝑠²𝜑_𝑥𝑥𝜑_𝑥𝑥𝑥𝑞²𝑑𝑥𝑑𝑡−2_𝑄𝑠⁴𝜑³_𝑥𝜑_𝑥𝑥𝑞²−3𝑑𝑥𝑑𝑡2_𝑄𝑠³𝜑²_𝑥𝜑_𝑥𝑥𝑞²𝑑𝑥𝑑𝑡+7_𝑄𝑠³𝜑_𝑥𝜑_𝑥𝑥𝜑_𝑥𝑥𝑥𝑞²𝑑𝑥𝑑𝑡+27_𝑄𝑠⁴𝜑²_𝑥𝜑²_𝑥𝑥𝑞²+3𝑑𝑥𝑑𝑡2_𝑄𝑠⁴𝜑³_𝑥𝜑_𝑥𝑥𝑥𝑞²𝑑𝑥𝑑𝑡+6_𝑄𝑠⁴𝜑³_𝑥𝜑_𝑥𝑥𝑥𝑞²𝑑𝑥𝑑𝑡−6_𝑄𝑠²𝜑_𝑥𝜑_𝑥𝑥𝑞²_𝑥𝑑𝑥𝑑𝑡+3_𝑄𝑠²𝜑_𝑥𝑥𝜑²_𝑥𝑥𝑥𝑞²3𝑑𝑥𝑑𝑡+2_𝑄𝑠²𝜑_𝑥𝑥𝜑_𝑥𝑥𝑥𝑞²𝑑𝑥𝑑𝑡+9_𝑄𝑠²𝜑_𝑥𝑥𝜑_𝑥𝑥𝑥𝑞²_𝑥𝑑𝑥𝑑𝑡+6_𝑄𝑠³𝜑_𝑥𝜑²_𝑥𝑥𝑞²9𝑑𝑥𝑑𝑡−2_𝑄𝑠⁴𝜑³_𝑥𝜑_𝑥𝑥𝑥𝑞²𝑑𝑥𝑑𝑡−272_𝑄𝑠⁴𝜑²_𝑥𝜑_𝑥𝑥𝑞²−𝑑𝑥𝑑𝑡_𝑄𝑠²𝜑_𝑥𝑥𝜑_𝑥𝑥𝑥𝑞²𝑑𝑥𝑑𝑡−3_𝑄𝑠²𝜑_𝑥𝜑_𝑥𝑥𝑥𝑞²_𝑥𝑑𝑥𝑑𝑡. (3.13) By the time derivative definition 𝜑 we can write 𝜑_𝑡≤𝐶𝜑²,𝜑_𝑡𝑡≤𝐶𝜑³, (3.14) (where 𝐶 does not depend on 𝑠,𝑡,𝑥). Using (3.14) for 𝑠 is sufficiently large and observes the same powers, then 𝐷₂≤𝐶_𝑄𝑠⁴𝜑²_𝑥𝜑²_𝑥𝑥𝑞²𝑑𝑥𝑑𝑡+_𝑄𝑠²𝜑_𝑥𝜑_𝑥𝑥𝑞_𝑥²𝑑𝑥𝑑𝑡. (3.15)
Now multiplying (3.7) by 𝑠𝜑𝑞|∇𝜓|² and integrating over 𝑄, we obtain _𝑄𝑠𝜑𝑞∇𝜓²𝐿₁𝑞𝑑𝑥𝑑𝑡+_𝑄𝑠𝜑𝑞∇𝜓²𝐿₂𝑞𝑑𝑥𝑑𝑡=_𝑄𝑠𝜑𝑞∇𝜓²ℎ_𝑠𝑑𝑥𝑑𝑡. (3.16) Therefore, _𝑄𝑠𝜑𝑞∇𝜓²ℎ_𝑠𝑑𝑥𝑑𝑡=_𝑄𝑠𝜑𝑞∇𝜓²𝐿₁𝑞𝑑𝑥𝑑𝑡+_𝑄𝑠𝜑∇𝜓²𝑞𝑞_𝑥𝑥+𝑑𝑥𝑑𝑡_𝑄𝑠𝜑∇𝜓²𝑞𝑞_𝑥𝑥𝑥𝑑𝑥𝑑𝑡−2_𝑄𝑠²𝜑𝜑_𝑥∇𝜓²𝑞𝑞_𝑥𝑑𝑥𝑑𝑡+3_𝑄𝑠³𝜑∇²_𝑥𝜓²𝑞𝑞_𝑥𝑑𝑥𝑑𝑡−3_𝑄𝑠²𝜑𝜑_𝑥𝑥∇𝜓²𝑞𝑞_𝑥𝑑𝑥𝑑𝑡−3_𝑄𝑠²𝜑𝜑_𝑥∇𝜓²𝑞𝑞_𝑥𝑥𝑑𝑥𝑑𝑡,_𝑄𝑠𝜑𝑞∇𝜓²𝐿₁1𝑞𝑑𝑥𝑑𝑡≤2_𝑄𝑠²𝜑²∇𝜓⁴𝑞²𝑑𝑥𝑑𝑡+2_𝑄𝐿₁𝑞²𝑑𝑥𝑑𝑡,_𝑄𝑠𝜑∇𝜓²𝑞𝑞_𝑥𝑥𝑑𝑥𝑑𝑡=−_𝑄𝑠𝜑∇𝜓²𝑞²_𝑥1𝑑𝑥𝑑𝑡+2_𝑄𝑠𝜑_𝑥𝑥∇𝜓²𝑞²+𝑑𝑥𝑑𝑡_𝑄𝑠𝜑_𝑥Δ𝜓∇𝜓𝑞²𝑑𝑥𝑑𝑡,_𝑄𝑠𝜑∇𝜓²𝑞𝑞_𝑥𝑥𝑥𝑑𝑥𝑑𝑡=_𝑄𝑠𝜑_𝑥∇𝜓²𝑞²_𝑥1𝑑𝑥𝑑𝑡+2_𝑄𝑠𝜑_𝑥𝑥𝑥∇𝜓²𝑞²+𝑑𝑥𝑑𝑡_𝑄𝑠𝜑_𝑥𝑥∇𝜓Δ𝜓𝑞²𝑑𝑥𝑑𝑡+2_𝑄𝑠𝜑∇𝜓²𝑞²_𝑥−𝑑𝑥𝑑𝑡_𝑄𝑠𝜑_𝑥∇𝜓²𝑞²𝑑𝑥𝑑𝑡−_𝑄𝑠𝜑∇𝜓²𝑞_𝑥𝑞_𝑥𝑥𝑑𝑥𝑑𝑡,−2_𝑄𝑠²𝜑_𝑥𝜑∇𝜓²𝑞𝑞_𝑥𝑑𝑥𝑑𝑡=_𝑄𝑠²𝜑²_𝑥∇𝜓²𝑞²𝑑𝑥𝑑𝑡+_𝑄𝑠²𝜑_𝑥𝑥𝜑∇𝜓²𝑞²𝑑𝑥𝑑𝑡+2_𝑄𝑠²𝜑_𝑥𝜑Δ𝜓∇𝜓𝑞²3𝑑𝑥𝑑𝑡,_𝑄𝑠³𝜑𝜑²_𝑥∇𝜓²𝑞𝑞_𝑥𝑑𝑥𝑑𝑡≤3_𝑄𝑠³𝜑²_𝑥𝜑∇𝜓⁴𝑞²3𝑑𝑥𝑑𝑡+4_𝑄𝑠³𝜑𝜑²_𝑥𝑞²_𝑥𝑑𝑥𝑑𝑡,−3_𝑄𝑠²𝜑𝜑_𝑥𝑥∇𝜓²𝑞𝑞_𝑥3𝑑𝑥𝑑𝑡=2_𝑄𝑠²𝜑_𝑥𝜑_𝑥𝑥∇𝜓²𝑞²3𝑑𝑥𝑑𝑡+2_𝑄𝑠²𝜑𝜑_𝑥𝑥𝑥∇𝜓²𝑞²𝑑𝑥𝑑𝑡+3_𝑄𝑠²𝜑𝜑_𝑥𝑥∇𝜓Δ𝜓𝑞²𝑑𝑥𝑑𝑡−3_𝑄𝑠²𝜑_𝑥𝜑∇𝜓²𝑞𝑞_𝑥𝑥𝑑𝑥𝑑𝑡=3_𝑄𝑠²𝜑𝜑_𝑥∇𝜓²𝑞²_𝑥𝑑𝑥𝑑𝑡−3_𝑄𝑠²𝜑_𝑥𝜑_𝑥𝑥∇𝜓²𝑞²𝑑𝑥𝑑𝑡−3_𝑄𝑠²𝜑²_𝑥∇𝜓Δ𝜓𝑞²3𝑑𝑥𝑑𝑡−2_𝑄𝑠²𝜑_𝑥𝜑_𝑥𝑥∇𝜓²𝑞²−3𝑑𝑥𝑑𝑡2_𝑄𝑠²𝜑𝜑_𝑥𝑥𝑥∇𝜓²𝑞²𝑑𝑥𝑑𝑡−3_𝑄𝑠²𝜑𝜑_𝑥𝑥∇𝜓Δ𝜓𝑞²−𝑑𝑥𝑑𝑡_𝑄𝑠²𝜑²_𝑥∇𝜓Δ𝜓𝑞²𝑑𝑥𝑑𝑡. (3.17) By using the above calculation in (3.16), it becomes −34_𝑄𝑠³𝜑𝜑²_𝑥𝑞²_𝑥𝑑𝑥𝑑𝑡=−_𝑄𝑠𝜑∇𝜓²𝑞ℎ_𝑠𝑑𝑥𝑑𝑡+_𝑄𝑠²𝜑²∇𝜓⁴𝑞²+𝑑𝑥𝑑𝑡_𝑄𝑠𝜑∇𝜓²𝑞²_𝑥1𝑑𝑥𝑑𝑡+4_𝑄𝐿₁𝑞²𝑑𝑥𝑑𝑡+2_𝑄𝑠𝜑Δ𝜓∇𝜓𝑞²_𝑥𝑑𝑥𝑑𝑡−_𝑄𝑠𝜑∇𝜓²𝑞_𝑥𝑞_𝑥𝑥𝑑𝑥𝑑𝑡+3_𝑄𝑠³𝜑𝜑_𝑥𝜑_𝑥𝑥∇𝜓²𝑞²+3𝑑𝑥𝑑𝑡2_𝑄𝑠²𝜑_𝑥𝜑_𝑥𝑥∇𝜓²𝑞²𝑑𝑥𝑑𝑡+_𝑄𝑠²𝜑𝜑_𝑥𝑥∇𝜓²𝑞²𝑑𝑥𝑑𝑡+2_𝑄𝑠²𝜑𝜑_𝑥∇𝜓Δ𝜓𝑞²+3𝑑𝑥𝑑𝑡2_𝑄𝑠²𝜑²_𝑥∇∇𝜓²𝑞²3𝑑𝑥𝑑𝑡+2_𝑄𝑠²𝜑_𝑥𝜑_𝑥𝑥∇𝜓²𝑞²3𝑑𝑥𝑑𝑡+2_𝑄𝑠³𝜑𝜑²_𝑥∇𝜓⁴𝑞²+5𝑑𝑥𝑑𝑡2_𝑄𝑠²𝜑_𝑥𝜑_𝑥𝑥∇𝜓²𝑞²1𝑑𝑥𝑑𝑡+2_𝑄𝑠𝜑_𝑥∇𝜓²𝑞²𝑑𝑥𝑑𝑡+3_𝑄𝑠²𝜑𝜑_𝑥𝑥𝑥∇𝜓²𝑞²𝑑𝑥𝑑𝑡+3_𝑄𝑠²𝜑𝜑_𝑥∇𝜓²𝑞²_𝑥𝑑𝑥𝑑𝑡−3_𝑄𝑠²𝜑_𝑥𝜑_𝑥𝑥∇𝜓²𝑞²3𝑑𝑥𝑑𝑡−2_𝑄𝑠²𝜑_𝑥𝜑_𝑥𝑥∇𝜓²𝑞²−3𝑑𝑥𝑑𝑡2_𝑄𝑠²𝜑𝜑_𝑥𝑥𝑥∇𝜓²𝑞²𝑑𝑥𝑑𝑡−3_𝑄𝑠²𝜑𝜑_𝑥𝑥∇𝜓Δ𝜓𝑞²𝑑𝑥𝑑𝑡−3_𝑄𝑠²𝜑𝜑_𝑥∇∇𝜓²𝑞²_𝑥1𝑑𝑥𝑑𝑡+4_𝑄𝑠𝜑∇𝜓⁴𝑞²_𝑥𝑑𝑥𝑑𝑡. (3.18) From the above equation (the right hand side), we can observe the powers of s: −34_𝑄𝑠³𝜑𝜑²_𝑥𝑞²_𝑥𝑑𝑥𝑑𝑡=𝐷₃+3_𝑄𝑠³𝜑𝜑²_𝑥∇𝜓⁴𝑞²𝑑𝑥𝑑𝑡+_𝑄𝑠𝜑𝑞²_𝑥𝑥𝑑𝑥𝑑𝑡, (3.19) where 𝐷₃=_𝑄𝑠𝜑𝑞∇𝜓²ℎ_𝑠𝑑𝑥𝑑𝑡+_𝑄𝑠𝜑∇𝜓²𝑞𝐿₁𝑞𝑑𝑥𝑑𝑡+3_𝑄𝑠²𝜑𝜑_𝑥∇𝜓²𝑞²_𝑥+3𝑑𝑥𝑑𝑡2_𝑄𝑠²𝜑_𝑥𝜑_𝑥𝑥∇𝜓²𝑞²𝑑𝑥𝑑𝑡+_𝑄𝑠²𝜑𝜑_𝑥𝑥∇𝜓²𝑞²𝑑𝑥𝑑𝑡+2_𝑄𝑠²𝜑𝜑_𝑥∇𝜓Δ𝜓𝑞²𝑑𝑥𝑑𝑡−3_𝑄𝑠³𝜑𝜑²_𝑥∇𝜓Δ𝜓𝑞²+3𝑑𝑥𝑑𝑡2_𝑄𝑠²𝜑_𝑥𝜑_𝑥𝑥∇𝜓²𝑞²3𝑑𝑥𝑑𝑡+2_𝑄𝑠²𝜑𝜑_𝑥𝑥𝑥∇𝜓²𝑞²𝑑𝑥𝑑𝑡+4_𝑄𝑠²𝜑_𝑥𝜑_𝑥𝑥∇𝜓²𝑞²3𝑑𝑥𝑑𝑡−2_𝑄𝑠²𝜑_𝑥𝜑_𝑥𝑥∇𝜓²𝑞²−3𝑑𝑥𝑑𝑡2_𝑄𝑠²𝜑𝜑_𝑥𝑥𝑥∇𝜓²𝑞²𝑑𝑥𝑑𝑡−3_𝑄𝑠²𝜑𝜑_𝑥𝑥∇𝜓Δ𝜓𝑞²𝑑𝑥𝑑𝑡. (3.20) Applying Cauchy’s inequality (with 𝜖=1) for 𝐷₃, 𝐷₃≤14_𝑄𝐿₁𝑞²1𝑑𝑥𝑑𝑡+4_𝑄ℎ_𝑠²3𝑑𝑥𝑑𝑡+2_𝑄𝑠²𝜑_𝑥𝜑_𝑥𝑥∇𝜓²𝑞²𝑑𝑥𝑑𝑡+3_𝑄𝑠²𝜑𝜑²_𝑥∇𝜓²𝑞_𝑥²𝑑𝑥𝑑𝑡. (3.21) Therefore 𝐿₁𝑞,𝐿₂𝑞_𝐿2(𝑄)=𝐷₂+𝐷₃+32_𝑄𝑠⁵𝜑⁴_𝑥𝜑_𝑥𝑥𝑞²3𝑑𝑥𝑑𝑡+4_𝑄𝑠³𝜑²_𝑥𝜑_𝑥𝑥𝑞²_𝑥𝑑𝑥𝑑𝑡+3_𝑄𝑠³𝜑𝜑²_𝑥∇𝜓⁴𝑞²𝑑𝑥𝑑𝑡+_𝑄𝑠𝜑𝑞²_𝑥𝑥𝑑𝑥𝑑𝑡. (3.22) By using (3.22) in (3.7), we get 𝐿₁𝑞²+𝐿₂𝑞²+3_𝑄𝑠⁵𝜑⁴_𝑥𝜑_𝑥𝑥𝑞²3𝑑𝑥𝑑𝑡+2_𝑄𝑠³𝜑²_𝑥𝜑_𝑥𝑥𝑞²_𝑥𝑑𝑥𝑑𝑡+2_𝑄𝑠𝜑𝑞²_𝑥𝑥𝑑𝑥𝑑𝑡≤𝐶𝑒^2𝑠𝜑_𝑄𝑔²𝑑𝑥𝑑𝑡+_𝑄^𝑇_𝑡𝑘(𝜏,𝑡)𝑒^−𝑠𝜑𝑞_𝑥𝑥𝑑𝜏²+𝑑𝑥𝑑𝑡_𝑄𝑠⁴𝜑²_𝑥𝜑²_𝑥𝑥𝑞²𝑑𝑥𝑑𝑡+_𝑄𝑠²𝜑_𝑥𝜑_𝑥𝑥𝑞_𝑥²+1𝑑𝑥𝑑𝑡2_𝑄𝐿₁𝑞²𝑑𝑥𝑑𝑡+6_𝑄𝑠³𝜑𝜑²_𝑥∇𝜓⁴𝑞²𝑑𝑥𝑑𝑡+6_𝑄𝑠²𝜑𝜑_𝑥∇𝜓²𝑞_𝑥²𝑑𝑥𝑑𝑡. (3.23) Recalling that 𝑠 is sufficiently large and also observing that powers of 𝑠 and |∇𝜓|>0 in Ω⧵𝜔, 12𝐿₁𝑞²+𝐿₂𝑞²+3_𝑄𝑠⁵𝜑⁴_𝑥𝜑_𝑥𝑥𝑞²3𝑑𝑥𝑑𝑡+2_𝑄𝑠³𝜑²_𝑥𝜑_𝑥𝑥𝑞²_𝑥𝑑𝑥𝑑𝑡+_𝑄𝑠𝜑𝑞²_𝑥𝑥𝑑𝑥𝑑𝑡≤𝐶_𝑄𝑒^2𝑠𝜑𝑔²𝑑𝑥𝑑𝑡+_𝑄𝑒^{2𝑠𝜑𝑇}_𝑡𝑘(𝜏,𝑡)𝑒^−𝑠𝜑𝑞_𝑥𝑥𝑑𝜏²+𝑑𝑥𝑑𝑡_𝑄𝜔𝑠⁴𝜑²_𝑥𝜑²_𝑥𝑥𝑞²𝑑𝑥𝑑𝑡+_𝑄𝜔𝑠²𝜑𝜑_𝑥𝑞_𝑥²𝑑𝑥𝑑𝑡. (3.24) From the above inequality 𝑞_𝑥 is in 𝜔₀,whichwill be eliminated (because 𝜔₀ is subset of 𝜔₀). To prove 𝑞_𝑥 as 𝑞 in 𝜔₀ (term |𝑞_𝑥|² on the right hand side of (3.24)), the truncating function is necessary, so let us define the truncating function 𝜌∈𝐶^∞₀(Ω) with 𝜌(𝑥)=1𝑖𝑛𝜔₀ and 𝜌(𝑥)=0inΩ⧵𝜔. Now we will multiply (3.7) by 𝜌𝑠𝜑𝑞 and integrating over 𝑄, then we have _𝑄𝜌𝑠𝜑𝑞𝐿₁𝑞𝑑𝑥𝑑𝑡+_𝑄𝜌𝑠𝜑𝑞𝐿₂𝑞𝑑𝑥𝑑𝑡=_𝑄𝜌𝑞𝑠𝜑ℎ_𝑠𝑑𝑥𝑑𝑡. (3.25) Since 𝜌_𝑄𝑠𝜑𝑞𝑞_𝑥𝑥𝑑𝑥𝑑𝑡=_𝑄𝜌𝑠𝜑_𝑥𝑥𝑞²𝑑𝑥𝑑𝑡−_𝑄𝜌𝑠𝜑_𝑥𝑞²_𝑥𝑑𝑥𝑑𝑡,_𝑄𝜌𝑠𝜑𝑞𝑞_𝑥𝑥𝑥𝑑𝑥𝑑𝑡=_𝑄𝜌𝑠𝜑_𝑥𝑞²_𝑥𝑑𝑥𝑑𝑡−_Σ𝑠𝜑𝜕𝑞𝜕𝑣²𝑑Σ+_𝑄𝜌𝑠𝜑_𝑥𝑞²_𝑥−𝑑𝑥𝑑𝑡_𝑄𝜌𝑠𝜑_𝑥𝑥𝑥𝑞²𝑑𝑥𝑑𝑡−3_𝑄𝜌𝑠²𝜑𝜑_𝑥𝑞𝑞_𝑥𝑥𝑑𝑥𝑑𝑡=3_𝑄𝜌𝑠²𝜑𝜑_𝑥𝑞²_𝑥𝑑𝑥𝑑𝑡−3_𝑄𝜌𝑠²𝜑_𝑥𝜑_𝑥𝑥𝑞²+3𝑑𝑥𝑑𝑡2_𝑄𝜌𝑠²𝜑𝜑_𝑥𝑥𝑥𝑞²𝑑𝑥𝑑𝑡−2_𝑄𝜌𝑠²𝜑_𝑥𝑞𝑞_𝑥𝑑𝑥𝑑𝑡=_𝑄𝜌𝑠²𝜑_𝑥𝑥𝜑𝑞²𝑑𝑥𝑑𝑡+_𝑄𝜌𝑠²𝜑²_𝑥𝑞²𝑑𝑥𝑑𝑡−3_𝑄𝜌𝑠²𝜑𝜑_𝑥𝑥𝑞𝑞_𝑥3𝑑𝑥𝑑𝑡=2_𝑄𝜌𝑠²𝜑_𝑥𝜑_𝑥𝑥𝑞²3𝑑𝑥𝑑𝑡−2_𝑄𝜌𝑠²𝜑𝜑_𝑥𝑥𝑥𝑞²3𝑑𝑥𝑑𝑡_𝑄𝜌𝑠³𝜑𝜑²_𝑥𝑞𝑞_𝑥3𝑑𝑥𝑑𝑡=−2_𝑄𝜌𝑠³𝜑³_𝑥𝑞²𝑑𝑥𝑑𝑡−3_𝑄𝜌𝑠³𝜑𝜑_𝑥𝜑_𝑥𝑥𝑞²𝑑𝑥𝑑𝑡. (3.26) The boundary term −∫_Σ𝑠𝜑|𝜕𝑞/𝜕𝑣|²𝑑Σ≥0, then _𝑄𝜌𝑠²𝜑𝜑_𝑥𝑞_𝑥²≤1𝑑𝑥𝑑𝑡4_𝑄𝑒^2𝑠𝜑𝑔²3𝑑𝑥𝑑𝑡−2_𝑄𝜌𝑠²𝜑𝜑_𝑥𝑥𝑞²3𝑑𝑥𝑑𝑡+2_𝑄𝜌𝑠²𝜑𝜑_𝑥𝑥𝑥𝑞²+𝑑𝑥𝑑𝑡_𝑄𝜌𝑠²𝜑𝜑_𝑡𝑞²𝑑𝑥𝑑𝑡+3_𝑄𝜌𝑠²𝜑_𝑥𝜑_𝑥𝑥𝑞²𝑑𝑥𝑑𝑡+_𝑄𝜌𝑠²𝜑𝜑_𝑥𝑥𝑥𝑞²+𝑑𝑥𝑑𝑡_𝑄𝜌𝑠𝜑_𝑡𝑞²𝑑𝑥𝑑𝑡−_𝑄𝜌𝑠³𝜑𝜑²_𝑥𝑞²𝑑𝑥𝑑𝑡+_𝑄𝜌𝑠⁴𝜑𝜑³_𝑥𝑞²𝑑𝑥𝑑𝑡+2_𝑄𝜌𝑠²𝜑²𝑞²1𝑑𝑥𝑑𝑡+4_𝑄𝜌𝑒^{2𝑠𝜑𝑇}_𝑡𝑘(𝜏,𝑡)𝑒^−𝑠𝜑𝑞_𝑥𝑥𝑑𝜏²𝑑𝑥𝑑𝑡. (3.27) Applying as 𝜌(𝑥)=1in𝜔₀⊂𝜔, _𝑄𝜔𝑠²𝜑𝜑_𝑥𝑞_𝑥²𝑑𝑥𝑑𝑡≤𝐶_𝑄𝑒^2𝑠𝜑𝑔²𝑑𝑥𝑑𝑡+_𝑄𝜔𝑒^{2𝑠𝜑𝑇}_𝑡𝑘(𝜏,𝑡)𝑒^−𝑠𝜑𝑞_𝑥𝑥𝑑𝜏²+𝑑𝑥𝑑𝑡_𝑄𝜔𝑠⁴𝜑𝜑³_𝑥𝑞²𝑑𝑥𝑑𝑡. (3.28) The term 𝑠,𝜑 is not in 𝜔, so it will be eliminated from the above inequalities. To eliminate 𝑠,𝜑, we will multiply (3.5) and (3.6) on (𝑠𝜑)⁻¹ (see [8, 12]): _𝑄𝑠𝜑⁻¹𝑞_𝑥²+𝑞_𝑥𝑥²+𝑞_𝑥𝑥𝑥²𝑑𝑥𝑑𝑡≤𝐶_𝑄𝑠𝜑⁻¹𝐿₂𝑞²𝑑𝑥𝑑𝑡+_𝑄𝑠𝜑⁻¹𝑠²𝜑_𝑥𝜑_𝑥𝑥𝑞²𝑑𝑥𝑑𝑡_𝑄𝑠𝜑⁻¹𝑞_𝑡²𝑑𝑥𝑑𝑡≤𝐶_𝑄𝑠𝜑⁻¹𝐿₁𝑞²𝑑𝑥𝑑𝑡+_𝑄𝑠⁴𝜑²_𝑥𝜑²_𝑥𝑥𝑞²𝑑𝑥𝑑𝑡+_𝑄𝑠²𝜑²_𝑡𝑞²𝑑𝑥𝑑𝑡 (3.29)(3.30) In view of the estimates (3.22)–(3.30), we obtain _𝑄𝑠𝜑⁻¹𝑞_𝑡²+𝑞_𝑥²+𝑞_𝑥𝑥²+𝑞_𝑥𝑥𝑥²+𝑑𝑥𝑑𝑡_𝑄𝑠⁵𝜑⁴_𝑥𝜑_𝑥𝑥𝑞²+𝑠³𝜑²_𝑥𝜑_𝑥𝑥𝑞_𝑥²+𝑠𝜑𝑞_𝑥𝑥²𝑑𝑥𝑑𝑡≤𝐶_𝑄𝑒^2𝑠𝜑𝑔²+_𝑄𝜔𝑠⁴𝜑²_𝑥𝜑²_𝑥𝑥+𝜑𝜑³_𝑥𝑞²+𝑑𝑥𝑑𝑡_𝑄𝜔𝑒^{2𝑠𝜑𝑇}_𝑡𝑘(𝜏,𝑡)𝑒^−𝑠𝜑𝑞_𝑥𝑥𝑑𝜏²𝑑𝑥𝑑𝑡. (3.31) From (3.31), we will eliminate the memory kernel term (that appears on the right hand side), because it may not contain in 𝜔. By using Hölder’s inequality and changing the order of integration, _𝑄𝜔𝑒^{2𝑠𝜑𝑇}_𝑡𝑘(𝜏,𝑡)𝑒^−𝑠𝜑𝑞_𝑥𝑥𝑑𝜏²≤𝑑𝑥𝑑𝑡_𝑄𝑒^{2𝑠𝜑𝑇}₀𝑘(𝜏,𝑡)𝑒^−𝑠𝜑𝑞_𝑥𝑥𝑑𝜏²≤𝑑𝑥𝑑𝑡_𝑄𝑒^{2𝑠𝜑𝑡2}_𝑡1|𝑘(𝜏,𝑡)|²𝑠𝜑𝑑𝜏^𝑡2_𝑡1𝑠⁻¹𝜑⁻¹𝑒^−𝑠𝜑𝑞_𝑥𝑥²‖𝑘‖𝑑𝜏𝑑𝑥𝑑𝑡≤𝐶²_𝐿^∞𝑒^2𝑠𝜑_{𝑄(𝑡1,𝑡2)×Ω}𝑒^2𝑠𝜑𝑠𝜑𝑒^−𝑠𝜑𝑞_𝑥𝑥²𝑑𝑥𝑑𝑡, (3.32) where 𝐶 depends on Ω,𝜔,𝑡₁,𝑡₂, and 𝑇 (since 𝑒^2𝑠𝜑𝑠⁻¹𝜑⁻¹≤𝐶<∞, and 𝑒^−2𝑠𝜑𝑠𝜑≤𝐶<∞ for all (𝑡,𝑥)∈((𝑡₁,𝑡₂)×Ω)). Finally reverting to the original variable 𝑤 to complete the proof, let us choose ∇𝑤=𝑒^−𝑠𝜑(𝑞_𝑥−𝑠𝜑_𝑥𝑞), then _𝑄𝑠𝜑⁻¹𝑒^2𝑠𝜑|∇𝑤|²𝑑𝑥𝑑𝑡≤𝐶_𝑄𝑠𝜑⁻¹𝑞_𝑥²𝑑𝑥𝑑𝑡+_𝑄𝑠𝜑⁻¹𝑠𝜑_𝑥𝑞²𝑑𝑥𝑑𝑡_𝑄,𝑠𝜑⁻¹𝑒^2𝑠𝜑|Δ𝑤|²𝑑𝑥𝑑𝑡≤𝐶_𝑄𝑠𝜑⁻¹𝑞_𝑥𝑥²𝑑𝑥𝑑𝑡+_𝑄𝑠𝜑⁻¹𝑠²𝜑²_𝑥𝑞²𝑑𝑥𝑑𝑡_𝑄,𝑠𝜑⁻¹𝑒^2𝑠𝜑|∇⋅Δ𝑤|²𝑑𝑥𝑑𝑡≤𝐶_𝑄𝑠𝜑⁻¹𝑞_𝑥𝑥𝑥²𝑑𝑥𝑑𝑡+_𝑄𝑒^2𝑠𝜑𝑠⁴𝜑²_𝑥𝜑²_𝑥𝑥𝑞²𝑑𝑥𝑑𝑡. (3.33) Similarly 𝑤_𝑡=𝑒^−𝑠𝜑(𝑞_𝑡−𝑠𝜑_𝑡𝑞), _𝑄𝑠𝜑⁻¹𝑒^2𝑠𝜑𝑤_𝑡²𝑑𝑥𝑑𝑡≤𝐶_𝑄𝑠𝜑⁻¹𝑞_𝑡²𝑑𝑥𝑑𝑡+_𝑄𝑠𝜑⁻¹𝑠²𝜑²_𝑡𝑞²𝑑𝑥𝑑𝑡. (3.34) From (3.32), we can observe the kernel term as _𝑄𝜔𝑒^{2𝑠𝜑𝑇}_𝑡𝑘(𝜏,𝑡)𝑒^−𝑠𝜑𝑞_𝑥𝑥𝑑𝜏²‖𝑘‖𝑑𝑥𝑑𝑡≤𝐶²_𝐿^∞_𝑄𝑒^2𝑠𝜑_{(𝑡1,𝑡2)×Ω}𝑤_𝑥𝑥²𝑑𝑥𝑑𝑡. (3.35) If 𝑠≥𝑠₀ is sufficiently large, then there exists a constant 𝐶>0 such that the above integral terms have been absorbed on 𝜔⊂Ω (the right hand side of (3.31)). From the identifier (3.33)–(3.35), we can conform all the terms |𝑤_𝑥𝑥𝑥|²,|𝑤_𝑡|²,|𝑤_𝑥𝑥|²,|𝑤|² involving in 𝑠≥𝑠₀,𝜔⊂Ω and we obtain (3.3).

Lemma 3.2. Suppose that Theorem 3.1 is satisfied. If there exist positive constants 𝐶,𝜇 (independent of 𝑠) and 𝑤 will be a solution of (2.1), such that the following inequality holds: _Ω|𝑤(0,𝑥)|²𝑑𝑥≤𝐶𝑒^𝜇𝑠_𝑄𝑒^2𝑠𝜑𝑔²𝑑𝑥𝑑𝑡+_𝑄𝜔𝑒^2𝑠𝜑𝜑²_𝑥𝜑²_𝑥𝑥|𝑤|²𝑑𝑥𝑑𝑡. (3.36)

Proof. We multiply (3.2) by 𝑤 and integrate on Ω (using Cauchy’s inequality, when 𝜖=1/2), we obtain −12𝑑𝑑𝑡_Ω|𝑤|²𝑑𝑥+2_Ω|∇𝑤|²1𝑑𝑥+2_Ω|𝑤|²𝑑𝑥𝑑𝑡+2_Ω|∇⋅(Δ𝑤)|²≤𝑑𝑥𝑑𝑡_Ω2𝑔²+32|𝑤|²𝑑𝑥+2_Ω^𝑇_𝑡𝑘(𝜏,𝑡)𝑤_𝑥𝑥𝑑𝜏²𝑑𝑥. (3.37) Define 𝛾(𝑡)=sup𝑒^−2𝑠𝜑𝜑_𝑥⁻²(𝑥,𝑡)𝜑⁻²_𝑥𝑥(𝑥,𝑡)∶𝑥∈Ω≤𝐶𝑒^{𝜇𝑠/𝛽(𝑡)},𝜇=2𝑒^{2𝑚‖𝜓‖∞}. (3.38) Therefore, −12𝑑𝑑𝑡_Ω|𝑤|²𝑑𝑥≤𝐶_Ω𝑔²𝑑𝑥+_Ω^𝑇_𝑡𝑘(𝜏,𝑡)𝑤_𝑥𝑥𝑑𝜏²𝑑𝑥+_Ω|𝑤|²+𝑑𝑥_Ω|∇𝑤|²𝑑𝑥+_Ω|∇⋅(Δ𝑤)|²𝑑𝑥. (3.39) Integrating (3.39) on (0,𝑡), −^𝑡₀𝑑𝑑𝑡_Ω|𝑤|²𝑑𝑥𝑑𝑠≤𝐶^𝑡_0Ω𝑔²𝑑𝑥+_Ω^𝑇_𝑡𝑘(𝜏,𝑡)𝑤_𝑥𝑥𝑑𝜏²𝑑𝑥+_Ω|∇𝑤|²𝑑𝑥+_Ω𝑒^2𝑠𝜑𝛾(𝑡)𝜑²_𝑥𝜑²_𝑥𝑥|𝑤|²𝑑𝑥+_Ω|∇⋅(Δ𝑤)|²𝑑𝑥𝑑𝑠,for𝑡∈(0,𝑇). (3.40) Therefore, _Ω|𝑤(0,𝑥)|²𝑑𝑥≤𝐶_Ω𝑒^2𝑠𝜑𝜑²_𝑥𝜑²_𝑥𝑥|𝑤|²𝑑𝑥+_Ω𝑔²𝑑𝑥+_Ω^𝑇_𝑡𝑘(𝜏,𝑡)𝑤_𝑥𝑥𝑑𝜏²+𝑑𝑥_Ω|∇𝑤|²𝑑𝑥+_Ω|∇⋅(Δ𝑤)|²𝑑𝑥for𝑡∈(0,𝑇). (3.41) Now we fix 𝑡₁ and 𝑡₂ such as 0<𝑡₁<𝑡₂<𝑇, then integrating the above inequality: ^𝑡2_𝑡1Ω|𝑤(0,𝑥)|²𝑒^{−𝜇𝑠/(𝑡(𝑡−𝑇))}𝑑𝑥𝑑𝑡≤𝐶^𝑡2_𝑡1Ω𝑒^2𝑠𝜑𝜑²_𝑥𝜑²_𝑥𝑥|𝑤|²𝑑𝑥+_Ω𝑔²𝑑𝑥+_Ω|∇𝑤|²𝑑𝑥+_Ω^𝑇_𝑡𝑘(𝜏,𝑡)𝑤_𝑥𝑥𝑑𝜏²𝑑𝑥+_Ω|∇⋅(Δ𝑤)|²𝑑𝑥𝑑𝑡. (3.42) As based on (3.32), the kernel has been modified as ^𝑡2_𝑡1Ω^𝑇_𝑡𝑘(𝜏,𝑡)𝑤_𝑥𝑥𝑑𝜏²≤𝑑𝑥𝑑𝑡_{𝑇0Ω𝑡2𝑡1}|𝑘(𝜏,𝑡)|²𝑒^−2𝑠𝜑𝑠𝜑𝑑𝜏^𝑡2_𝑡1𝑤_𝑥𝑥²𝑒^2𝑠𝜑𝑠⁻¹𝜑⁻¹‖𝑘‖𝑑𝜏𝑑𝑥𝑑𝑡≤𝐶²_𝐿^∞𝑡2_𝑡1Ω𝑒^2𝑠𝜑𝑠⁻¹𝜑⁻¹𝑤_𝑥𝑥²𝑑𝑥𝑑𝑡. (3.43) Using (3.43) in (3.42), then ^𝑡2_𝑡1Ω|𝑤(0,𝑥)|²𝑒^{−𝜇𝑠/(𝑡(𝑡−𝑇))}𝑑𝑥𝑑𝑡≤𝐶_𝑄𝑔²𝑑𝑥𝑑𝑡+_𝑄|∇𝑤|²𝑑𝑥𝑑𝑡+_𝑄|∇⋅(Δ𝑤)|²𝑑𝑥𝑑𝑡+^𝑡2_𝑡1Ω𝑒^2𝑠𝜑𝜑²_𝑥𝜑²_𝑥𝑥|𝑤|²+𝑠⁻¹𝜑⁻¹𝑤_𝑥𝑥²𝑑𝑥𝑑𝑡. (3.44) Since inf_{𝑡𝑡∈1,𝑡2}𝑒^{−𝜇𝑠/𝛽(𝑡)}≥𝐶>0 (3.45) and the Carleman estimate (3.3) using in (3.44), we obtain _Ω|𝑤(0,𝑥)|²𝑑𝑥≤𝐶_𝑄𝑒^2𝑠𝜑𝑔²𝑑𝑥𝑑𝑡+_𝑄𝜔𝑒^2𝑠𝜑𝜑²_𝑥𝜑²_𝑥𝑥|𝑤|²𝑑𝑥𝑑𝑡. (3.46)

4. Controllability Results

Now we are ready to give the proof of Theorem 2.2 result, which will be a main part of our work.

Proof of Theorem 2.2. Let us fix 𝑇>0,𝑎∈𝑊^1,∞(0,𝑇∶𝐿²(𝜔)),𝑏∈𝐿^∞(𝑄),and 𝑦₀∈𝐻¹₀(Ω). For every 𝜖>0, the penalized formula Minimize𝐻_𝜖(𝑢);𝑢∈𝐿²(𝑄)>0, (4.1) where the functional 𝐻_𝜖 is (see [9, 11]) 𝐻_𝜖(𝑢)=12_𝑄|𝑢|²1𝑑𝑥𝑑𝑡+2𝜖_Ω𝑦(𝑇,𝑥)²𝑑𝑥. (4.2) Here 𝑦 is a solution of (1.2), which is associated with the control 𝑢. Since 𝐻_𝜖 is a continuous strictly convex functional in 𝐿²(𝑄), then 𝐻_𝜖 has a unique solution (𝑢_𝜖,𝑦_𝜖) (for any 𝜖>0). If (𝑢_𝜖,𝑦_𝜖) (as 𝜖→0) is a null controllability solution of the system (1.1), then due to penalization property (1/𝜖)∫_Ω|𝑦(𝑇,𝑥)|²𝑑𝑥, the limit exists in an appropriate norm. As based on the pontryagin maximum principle, the maximal condition on the control 𝑢_𝜖 is 𝑢_𝜖=𝜒_𝜔𝑤_𝜖𝑎.𝑒in𝑄, (4.3) and 𝑤^𝑢∈𝐶([0,𝑇];𝐿²(Ω))∩𝐿²(0,𝑇;𝐻¹₀(Ω)) is a solution for 𝑤_𝜖𝑡+Δ𝑤_𝜖+𝑤_𝜖𝑥−𝑤_𝜖+𝑤_{𝜖𝑥𝑥𝑥}−^𝑇_𝑡𝑘(𝜏,𝑡)𝑤_𝜖𝑥𝑥𝑤𝑑𝜏=0in𝑄,_𝜖1=−𝜖𝑦_𝜖()𝑤𝑇,𝑥,inΩ,_𝜖=0,inΣ. (4.4)(4.5)(4.6) Now we multiply (4.4) by 𝑦_𝜖 (where 𝑦=𝑦_𝜖), (1.2) by 𝑤_𝜖 and integrate on 𝑄, _𝑄𝜔𝑦_𝜖²1𝑑𝑥𝑑𝑡+𝜖_Ω𝑦_𝜖(𝑇,𝑥)²𝑑𝑥−_Ω𝑦₀(𝑥)𝑤_𝜖(𝑥,0)𝑑𝑥+_𝑄𝑓𝑤_𝜖𝑑𝑥𝑑𝑡=0. (4.7) Applying the observability inequality (3.36), we have _Ω𝑦₀(𝑥)𝑤_𝜖(𝑥,0)𝑑𝑥≤𝐶_𝑄𝜔𝑤_𝜖(𝑥,0)²𝑑𝑥𝑑𝑡^1/2𝑦₀₂. (4.8) Using the Carleman estimate (3.3) and Lemma 2.1 condition, _𝑄𝑓𝑤_𝜖𝑑𝑥𝑑𝑡≤𝐶_𝑄𝑒^2𝑠𝜑𝜑²_𝑥𝜑²_𝑥𝑥𝑤_𝜖²𝑑𝑥𝑑𝑡^1/2_𝑄𝑓²₁𝑑𝑥𝑑𝑡^1/2≤𝐶_𝑄𝜔𝑒^2𝑠𝜑𝜑²_𝑥𝜑²_𝑥𝑥𝑤_𝜖²𝑑𝑥𝑑𝑡^1/2𝑓_1𝐿2(Ω), (4.9) where 𝐶 is a positive constant that is independent of 𝜖 and 𝑦₀. By virtue of (4.7)–(4.9), we obtain _𝑄𝑦_𝜖²1𝑑𝑥𝑑𝑡+𝜖_Ω𝑦_𝜖(𝑇,𝑥)²𝑑𝑥≤𝐶𝑦₀²₂+𝑓₁²_𝐿2(𝑄),∀𝜖>0. (4.10) By the condition (4.10), we can observe _𝑄𝑢_𝜖²1𝑑𝑥𝑑𝑡+𝜖_Ω𝑦_𝜖(𝑇,𝑥)²𝑑𝑥≤𝐶𝑦₀²₂+𝑓₁²_𝐿2(𝑄). (4.11) Since 𝑢_𝜖 is bounded in 𝐿²(𝑄), there exists a subsequence denoted by 𝜖 such that 𝑢_𝜖→𝑢^∗weaklyin𝐿²(𝑄),𝑦_𝜖→𝑦^∗weaklyin𝐿²0,𝑇;𝐻¹(Ω)as𝜖→0. (4.12) Clearly 𝑦^∗=𝑦^𝑢∗, letting 𝜖→0 in (4.11), then we have |𝑦_𝜖(𝑇,𝑥)|²=0a.e.𝑥∈Ω. This completes the proof of Theorem 2.2.

5. Conclusions

In this paper, we investigated the null controllability result for linearized KdV-Burgers equation with memory. It is suggested as a model which permits an analytical treatment of the Carleman estimate to prove the stability. The observability inequality has observed the null controllability, and the kernel term expression been observed in 𝑦_𝑥𝑥. It grants the stability, so our model could be heuristically useful in the studies of nonlinear water waves in a shallow channel, compressible gas, strong hydrodynamic turbulence as well as pressure force, and so forth, [1, 2].

Acknowledgments

This research was partially supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MEST) (no. 2012-038978) and research funds of Chonbuk National University in 2011.