On the solvability of the neumann boundary value problem without initial conditions for hyperbolic systems in infinite cylinders

1. Introduction Motivated by the fact that abstract boundary value problems for hyperbolic systems arise in many areas of applied mathematics, this type of system has received considerable attention for many years (see [2, 6, 9, 10]). Naturally, when expanding from hyperbolic systems with initial conditions in cylinders (0, ∞) × Ω studied in [9], we consider one in infinite cylinders (−∞, +∞)×Ω, where Ω is a bounded domain in Rn with the boundary S = ∂Ω. Base on previous achievements and direction [8], we deal with the solvability of the Neumann boundary value problem without initial conditions for hyperbolic systems in infinite cylinders.

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JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci., 2013, Vol. 58, No. 7, pp. 3-14 This paper is available online at ON THE SOLVABILITY OF THE NEUMANN BOUNDARY VALUE PROBLEM WITHOUT INITIAL CONDITIONS FOR HYPERBOLIC SYSTEMS IN INFINITE CYLINDERS Nguyen Manh Hung1 and Nguyen Thi Van Anh2 1National Institute of Education Management 2Hanoi National University of Education Abstract. In this paper, we study the Neumann boundary value problem without initial conditions for hyperbolic systems in infinite cylinders. The primary results obtained are the recognition of the uniqueness and the existence of generalized solutions. Keywords Solvability, generalized solution, problems without initial conditions, gronwall Bellman inequality. 1. Introduction Motivated by the fact that abstract boundary value problems for hyperbolic systems arise in many areas of applied mathematics, this type of system has received considerable attention for many years (see [2, 6, 9, 10]). Naturally, when expanding from hyperbolic systems with initial conditions in cylinders (0,∞)× Ω studied in [9], we consider one in infinite cylinders (−∞,+∞)×Ω, where Ω is a bounded domain in Rn with the boundary S = ∂Ω. Base on previous achievements and direction [8], we deal with the solvability of the Neumann boundary value problem without initial conditions for hyperbolic systems in infinite cylinders. For a < b, set Qba = Ω × (a, b), Sba = S × (a, b). Let u = (u1, . . . , us) be a complex-valued vector function and let us introduce some functional spaces used throughout in this paper. Received September 20, 2013. Accepted October 30, 2013. Contact Nguyen Thi Van Anh, e-mail address: vananh89nb@gmail.com 3 Nguyen Manh Hung and Nguyen Thi Van Anh We use Hk,l(QR) the space consisting of all vector functions u : QR −→ Cs satisfying: ∥u∥2Hk;l(QR) := ∫ QR ( k∑ |α=0| |Dαu|2 + l∑ j=1 |utj |2 ) dxdt, and Hk,l(e−γt, QR) is the space of vector functions with norm ∥u∥2Hk;l(e− t,QR) := ∫ QR ( k∑ |α=0| |Dαu|2 + l∑ j=1 |utj |2 ) e−2γtdxdt. In particular ∥u∥2Hk;0(e− t,QR) := ∫ QR k∑ |α=0| |Dαu|2e−2γtdxdt. Especially, we set L2(e−γt, QR) = H0,0(e−γt, QR). We denote by Hk,l(e−γt, QR) = { u ∈ Hk,l(e−γt, QR) such that lim t→−∞ ∥u(., t)∥L2(Ω) = 0 } , soHk,l(e−γt, QR) is a linear space. Adding that, we consider an important spaceHk,l(eγt, QR) to be the space of vector functions with norm ∥u∥2Hk;l(e t,QR) := ∫ QR ( k∑ |α=0| |Dαu|2 + l∑ j=1 |utj |2 ) e2γtdxdt < +∞. In particular ∥u∥2Hk;0(e t,QR) := ∫ QR k∑ |α|=0 |Dαu|2e2γtdxdt < +∞. We introduce the matrix differential operator: L(x, t,D) = m∑ |p|,|q|=0 Dp(apq(x, t)D q), where the coefficients apq are s × s matrices of functions with bounded complex-valued components in QR, apq = (−1)|p|+|q|a∗qp, with a∗qp being complex conjugate transportation matrices of apq. 4 On the solvability of the neumann boundary value problem without initial conditions... We recall Green’s formula (Theorem 9.47, [5]): Let Bi(x,D), i = 1, ...,m, be a Dirichlet system of the order i − 1. Assume that Ω and the coefficients of the operators involved are sufficiently smooth. Then there exist normal boundary-value operators Ni, of order 2m− 1− ordBi. such that, for all u, v ∈ H2m(Ω), we have:∫ Ω m∑ |α|,|β|=0 aαβ(x)D βuDαvdx = ∫ Ω ( m∑ |α|,|β|=0 (−1)|α|Dα(aαβ(x)Dβu) ) vdx − ∫ ∂Ω m∑ j=1 (Bjv)(Nju)dS. We assume further that Ω and {apq} satisfy Green’s formula and we define Nj, j = 1, . . . ,m- the system of operators on the boundary SR. Denote by B(u, v)(t) = m∑ |p|,|q|=0 (−1)|p| ∫ Ω apqD quDpvdx, and HmN (Ω) = { u ∈ Hm(Ω) : Nju = 0 on S for all j = 1, . . . ,m } . We assume further that the form (−1)mB(., .)(t) is HmN - uniformly elliptic with respect to t and that means there exists a constant µ0 > 0 independent of t and u such that: (−1)mB(u, u)(t) ≥ µ0∥u(., t)∥2Hm(Ω) for all u ∈ HmN (Ω), and a.e t ≥ h. We consider the hyperbolic system in the cylinder QR (−1)m−1L(x, t,D)u− utt = f(x, t) in QR, (1.1) Nju SR = 0, j = 1, . . . ,m. (1.2) Definition 1.1. Let f ∈ L2(e−γt, QR), a complex-valued vector function u ∈ Hm,1(e−γt, QR) is called a generalized solution of problem (1.1) - (1.2) if and only if for any T > 0 the equality: (−1)m−1 ∫ T −∞ B(u, η)(t)dt+ ∫ QT−∞ utηtdxdt = ∫ QT−∞ fηdxdt, (1.3) holds for all η ∈ Hm,1(eγt, QR), η(x, t) = 0 with t ≥ T . 5 Nguyen Manh Hung and Nguyen Thi Van Anh 2. The uniqueness of a generalized solution of a problem (1.1) - (1.2) Theorem 2.1. If γ > 0 and ∂apq ∂t < µ1e2γt,∀t ∈ R, ∀|p|, |q| ≤ m, the problem (1.1)-(1.2) has no more than one solution. Proof. Assume u1(x, t) and u2(x, t) to be two generalized solutions of problem (1.1) - (1.2), set u(x, t) = u1(x, t)− u2(x, t). For any T > 0, b ≤ T , denote: u(x, t) = u1(x, t)− u2(x, t), η(x, t) =  t∫ b u(x, τ)dτ, −∞ ≤ t ≤ b, 0 , b ≤ t ≤ T. So we get η(x, T ) = 0, η(x, t) ∈ Hm,1(eγt, QT−∞), and ηt(x, t) = u(x, t), ∀(x, t) ∈ Qb−∞. Then we use η as a test function and because u = ηt, according to the definition of the generalized solution, we have: (−1)m−1 m∑ |p|,|q|=0 (−1)|p| ∫ Qb−∞ apqD qηtDpηdxdt+ ∫ Qb−∞ ηttηtdxdt = 0. (2.1) Adding the equation (2.1) with its complex conjugate we get: (−1)m−12Re ∫ b −∞ B(ηt, η)(t)dt+ 2Re ∫ Qb−∞ ηttηtdxdt = 0. (2.2) We transform the first term using integration by parts and the hypotheses of the coefficients and for the second term we use integration by parts, then replacing the obtained equalities into (2.2), we get: ∥ηt(., b)∥2L2(Ω)+ limh→−∞(−1) mB(η, η)(h) = (−1)m−1 m∑ |p|,|q|=0 (−1)|p| ∫ Qb−∞ ∂apq ∂t DqηDpηdxdt. Noting the asumption, we then have the fact that the coeficients apq are continuous with respect to the time variable and η ∈ Hm,1(eγt, QT−∞), so there exists the limit lim h→−∞ (−1)mB(η, η)(h). By using a uniformly elliptic condition we imply: lim h→−∞ (−1)mB(η, η)(h) ≥ µ0 lim h→−∞ ∥η(., h)∥2Hm(Ω), and thus ∥ηt(., b)∥2L2(Ω)+µ0 limh→−∞ ∥η(., h)∥ 2 Hm(Ω) ≤ (−1)m−1 m∑ |p|,|q|=0 (−1)|p| ∫ Qb−∞ ∂apq ∂t DqηDpηdxdt. 6 On the solvability of the neumann boundary value problem without initial conditions... By using the Cauchy inequality, we have: (−1)m−1 m∑ |p|,|q|=0 (−1)|p| ∫ Qb−∞ ∂apq ∂t DqηDpηdxdt ≤ µ1m∗ ∫ b −∞ ∥η(., t)∥2Hm(Ω)e2γtdt. We have yields: ∥ηt(., b)∥2L2(Ω) + µ0 limh→−∞ ∥η(., h)∥ 2 Hm(Ω) ≤ µ1m∗ ∫ b −∞ ∥η(., t)∥2Hm(Ω)e2γtdt. (2.3) Now denote by: vp (x, t) = ∫ h t Dpu(x, τ)dτ, −∞ ≤ t ≤ b. Hence, we can see that Dpη(x, t) = ∫ t b Dpu(x, τ)dτ = vp(x, b)− vp(x, t), lim h→−∞ Dpη(x, h) = vp(x, b), lim h→−∞ ∥η(., h)∥2Hm(Ω) = m∑ |p|=0 ∫ Ω |vp(x, b)|2dx, and from the equality (2.3) we have: ∥ηt(., b)∥2L2(Ω) + µ0 m∑ |p|=0 ∫ Ω |vp(x, b)|2dx ≤ µ1m∗ ∫ b −∞ ∥η(., t)∥2Hm(Ω)e2γtdt. This then leads to ∥ηt(., b)∥2L2(Ω) + µ0 m∑ |p|=0 ∥vp(x, b)∥2L2(Ω) ≤ µ1m∗ m∑ |p|=0 ∫ Qb−∞ e2γt|Dpη(x, t)|2dxdt ≤ 2µ1m∗e2γb m∑ |p|=0 ∥vp(., b)∥2L2(Ω) + 2µ1m∗ m∑ |p|=0 ∫ b −∞ e2γt∥vp(., t)∥2L2(Ω)dt =⇒∥ηt(., b)∥2L2(Ω) + (µ0 − 2µ1m∗e2γb) m∑ |p|=0 ∥vp(x, b)∥2L2(Ω) ≤ 2µ1m ∗ µ0 − 2µ1m∗e2γb ∫ b −∞ ( e2γt∥ηt(., t)∥2L2(Ω) + (µ0 − 2µ1m∗e2γb)e2γt m∑ |p|=0 ∥vp(x, b)∥2L2(Ω) ) dt. So, there exists a positive number C > 0, C > µ0 2µ1m∗ such that ∥ηt(., b)∥2L2(Ω) + m∑ |p|=0 ∥vp(x, b)∥2L2(Ω) ≤ C ∫ b −∞ e2γt (∥ηt(., t)∥2L2(Ω) + m∑ |p|=0 ∥vp(x, b)∥2L2(Ω) ) dt. 7 Nguyen Manh Hung and Nguyen Thi Van Anh Put J(t) = ∥ηt(x, t)∥2L2(Ω) + (µ0 − 2µ1m∗e2γb) m∑ |p|=0 ∥vp(x, t)∥2L2(Ω), we have: J(b) ≤ C ∫ b −∞ e2γtJ(t)dt, for a.e b ≤ 1 2γ ln 1 C . By performing a check similar to the proof of the Gronwall- Bellman inequality (see [4], page 624-625), we will prove that J(t) ≡ 0 on (−∞, 1 2γ ln 1 C ]. In fact, taking ζ(t) = ∫ t −∞ e 2γsJ(s)ds, we have ζ ′(t) = ( ∫ 0 −∞ e 2γsJ(s)ds +∫ t 0 e2γsJ(s)ds)′ = e2γtJ(t), then we have: ζ ′(t) ≤ Ce2γtζ(t) for a.e t ≤ 1 2γ ln 1 C . From this we see d ds ( ζ(s)e −Ce2 s 2 ) = e −Ce2 s 2 (ζ ′(s)− Ce2γtζ(t)) ≤ 0. By integrating with respect to s from −∞ to t in remark that lim s→−∞ ζ(s)e −Ce2 s 2 = 0, we get ζ(t)e −Ce2 t 2 ≤ 0 for a.e t ≤ 1 2γ ln 1 C . Thus we obtain ζ(t) ≤ 0 and we can conclude ζ ′(t) ≤ 0 for a.e t ≤ 1 2γ ln 1 C by the above estimate. From this, one has the desied estimate. So u(x, t) = 0 almost everywhere t ∈ (−∞, 1 2γ ln 1 C ]. Because of the uniqueness of the solution of a problem with initial conditions for a hyperbolic system, we imply u1(x, t) = u2(x, t) almost everywhere t ∈ R. We note that the obtained result about the uniqueness does not change if we consider the partial differential equations in the forms: (−1)mLu− utt − αut = f, (x, t) ∈ QR, (i) Nju SR = 0, j = 1, 2, . . . ,m. (ii) where α is a positive constant number. We have the definition of generalized solutions of the problem (i) - (ii) 8 On the solvability of the neumann boundary value problem without initial conditions... Definition 2.1. Let f ∈ L2(e−γt, QR), a complex-valued vector function u ∈ Hm,1(e−γt, QR) is called a generalization of problem (i)-(ii) if and only if for any T > 0 the equality: (−1)m−1 ∫ T −∞ B(u, η)(t)dt+ ∫ QR ut(ηt − αη)dxdt = ∫ QT−∞ fηdxdt (iii) holds for all η ∈ Hm,1(eγt, QR), η(x, t) = 0 with t ≥ T . By the same proofs we give the theorem about the uniqueness of this problem. Theorem 2.2. If γ > 0 and ∂apq ∂t < µ1e2γt,∀t ∈ R,∀|p|, |q| ≤ m, the problem (i) - (ii) has only one solution. 3. The existence of a generalized solution of a Neumann boundary value problem for hyperbolic system with initial conditions First, we set the hyperbolic systems (1.1) - (1.2) in Q∞h with initial conditions u(x, h) = ut(x, h) = 0. (1.3 ′) We restate the concept of the generalized solution of (1.1)-(1.2)-(1.3’). A function u(x, t) is called a generalized solution of the problem (1.1)-(1.2)-(1.3’) in the space Hm,1(e−γt, Q∞h ), if and only if u(x, t) belongs to H m,1(e−γt, Q∞h ), u(x, h) = 0, and the equality (−1)m−1 ∫ QTh m∑ |p|,|q|=0 (−1)|p|apqDquDpη + ∫ QTh utηtdxdt = ∫ QTh fηdxdt holds for all η belong to Hm,1(QTh ) satisfying η(x, T ) = 0, for all T > h. Theorem 3.1. (The existence of a generalized solution) Let γ > γ0 = µm ∗ 2µ0 , here m∗ = m∑ |p|=0 1. Assume that the operator (−1)mB(., .)(t) satisfies the elliptic uniformity condition and (i) sup { ∂apq ∂t , |apq| : (x, t) ∈ Q∞h , 0 ≤ |p|, |q| ≤ m} ≤ µ, (ii) f(x, t) ∈ L2(e−γt, Q∞h ) , 9 Nguyen Manh Hung and Nguyen Thi Van Anh Then there exists a unique generalized solution u(x, t) ∈ Hm,1(e−γt, Q∞h ) of problem (1.1) - (1.2) - (1.3’) satisfying: ∥u∥2Hm;1(e− t,Q∞h ) ≤ C∥f∥ 2 L2 ( e− t,Q∞h ), (3.1) where C = const > 0 is independent of u, h and f . Proof. The uniqueness is similar way to that in [9]. We omit the details here. Note that the constant C in the estimate in Theorem 2.1 in [9] depends on t = 0, so if we change the initial conditions by t = h in the same proof, we also obtain the fact that the constant C is depentdent on t = h. Now we give the proof to improve it. Due to the similarities as in [3], we get the approximate solutions {uN(x, t)}∞N=1 defined that uN(x, t) = ∑N k=1 c N k (t)φk(x) such that c N l (h) = 0 and d dt cNl (h) = 0, l = 1, ..., N and (−1)m−1 m∑ |p|,|q|=0 (−1)|p| ∫ Ω apqD quNDpφldx− ∫ Ω uNttφldx = ∫ Ω fφldx. (3.2) Multiplying (3.2) by dCNl (t) dt and taking the sum with respect from 1 to N , integrating with respect to t from h to τ ( τ ≥ h), then adding that to its complex conjugate and finally applying apq = (−1)|p|+|q|a∗qp, from the initial conditions of uN we conclude that ∥uNt (., τ)∥2L2(Ω) + (−1)mB(uN , uN)(τ) = −(−1)m−1 m∑ |p|,|q|=0 (−1)|p| ∫ Qh ∂apq ∂t DquNDpuNdxdt− 2Re ∫ Qh fuNt dxdt, From the uniformly elliptic condition of the operator (−1)mB(., .)(t) and the bounded property of the functions apq, ∂apq ∂t , and the Cauchy inequality we get ∥uNt (., τ)∥2L2(Ω) + µ0∥uN(., τ)∥2Hm(Ω) ≤ δ ∫ τ h (∥uNt (., t)∥2L2(Ω) + µm∗δ ∥uN(., t)∥2Hm(Ω))dt+ 1δ ∫ τ h ∥f(., t)∥2L2(Ω)dt. We take µm ∗ δ = µ0 then ∥uNt (., τ)∥2L2(Ω) + µ0∥uN(., τ)∥2Hm(Ω) ≤ 2γ0 ∫ τ h (∥uNt (., t)∥2L2(Ω) + µ0∥uN(., t)∥2Hm(Ω))dt+ C ∫ τ h ∥f(., t)∥2L2(Ω)dt. Set JN(t) = ∥uNt (., τ)∥2L2(Ω) + µ0∥uN(., τ)∥2Hm(Ω), 10 On the solvability of the neumann boundary value problem without initial conditions... we get JN(τ) ≤ 2γ0 ∫ τ h JN(t)dt+ C ∫ τ h ∥f(., t)∥2L2(Ω)dt. Using the Gronwall-Bellman inequality we have: JN(τ) ≤ C τ∫ h ∥f(., t)∥2L2(Ω)dt+ 2γ0 τ∫ h e2γ0(τ−s) s∫ h ∥f(., θ)∥2L2(Ω)dθds. So, the following inequality is obvious ∥uNt (., τ)∥2L2(Ω) + ∥uN(., τ)∥2Hm(Ω) (3.3) ≤ C1 τ∫ h ∥f(., t)∥2L2(Ω)dt+ C22γ0 τ∫ h e2γ0(τ−s) s∫ h ∥f(., θ)∥2L2(Ω)dθds, with the constant C1, C2 not depending on h. Now multiplying both sides of this inequality by e−2γt. Then integrating with respect to τ from h to∞ we have:∫ ∞ h ( ∥uNt (., τ)∥2L2(Ω) + ∥uN(., τ)∥2Hm(Ω) ) e−2γtdτ ≤ C1 +∞∫ h e−2γτ τ∫ h ∥f(., t)∥2L2(Ω)dtdτ + C2 +∞∫ h e−2γτ τ∫ h e2γ0(τ−s) s∫ h ∥f(., θ)∥2L2(Ω)dθdsdτ. (3.4) Denote by I1, I2 the terms from the first and second respectively of the right-hand sides of above inequalty. We will give estimations for these terms. First, I1 = C1 +∞∫ h ∥f(., t)∥2L2(Ω) +∞∫ t e−2γτdτdt = C1 2γ ∥f∥2L2(e− t,Q∞h ), and I2 = C2 +∞∫ h ∥f(., θ)∥2L2(Ω) +∞∫ θ e−2γ0s +∞∫ s e2(γ0−γ)τdτdsdθ = C2 4γ(γ − γ0)∥f∥ 2 L2(e− t,Q∞h ) , here C1, C2 are constants not depending on h, f , uN . Combining the above estimate we get: ∥uN∥2Hm;1(e− t,Q∞h ) ≤ C∥f∥ 2 L2(e− t,Q∞h ) . From this, in the same manner as in Theorem 2.1 (see [9]), we can conclude that there exists a generalized solution of the problem satisfying (3.1). 11 Nguyen Manh Hung and Nguyen Thi Van Anh 4. The existence of a generalized solution of the problem (1.1) - (1.2) A generalized solution of a problem (1.1) - (1.2) can be approximated by a sequence of solutions of problems with initial conditions (1.1) - (1.2) - (1.3’) in cylinder Q∞h . Consider in the real line R, we use Theory 5.5 ([5]), we then see that there exists a test function θ ∈ C∞(R) such that θ(t) = 0,∀t ≤ 0, θ(t) = 1∀t ≥ 1, θ(t) ∈ [0, 1],∀t ∈ [0, 1]. Moreover, we can suppose that all derivatives of θ(t) are bounded. Let h ∈ (−∞, 0] be an integer. Set fh(x, t) = θ(t− h)f(x, t) =⇒ fh(x, t) ∈ L2(e−γt, QR), h ∈ Z. fh(., t) = { f(., t) if t ≥ h+ 1 0 if t < h And we have ∥fh∥2L2(e− t,R) ≤ ∥f∥2L2(e− t,QR) We consider the following problem in the cylinder Q∞h : (−1)m−1Lu− utt = fh in Q∞h , Nju = 0, j = 1, . . . ,m on S∞h , u t=h = ut t=h = 0 on Ω. It is easy to see that there exists a number γ0 > 0 such that for each γ > γ0 the above problem has a unique generalized solution called uh in Hm,1(e−γt, Q∞h ) which the following estimate satisfies: ∥uh∥2Hm;1(e− t,QR) ≤ C∥f∥2L2(e− t,QR). Let consider genenalized solutions uh and uk of problems in cylinders Q∞h and Q∞k with f(x, t) is replaced by f h(x, t) and fk(x, t) respectively. If h > k, uh can be understood in Hm,1(e−γt, Q∞k ) with u h(x, t) = 0, ∀k ≤ t ≤ h. Setting vkh = uk − uh, fkh = fk − fh, so vkh is the generalized solution of the following problem: (−1)m−1Lv − vtt = fkh in Q∞k , Njv = 0, j = 1, . . . ,m on S∞k , v t=k = vt t=k = 0 on Ω. Then, we get ∥ukh∥2Hm;1(e− t,QR) ≤ C∥fkh∥2L2(e− t,QR) From the definition of fh we can see that {fh}−∞h=0 is the Cauchy sequence in the space L2(e−γt, QR), it follows that {uh}−∞h=0 is a Cauchy sequence in the completed space Hm,1(e−γt, QR) and thus, uh converge to u in Hm,1(e−γt, QR). 12 On the solvability of the neumann boundary value problem without initial conditions... On the other hand, as uh(x, t) = 0, t ≤ h we can take u satisfying limh→−∞ ∥u(., h)∥2L2(Ω) = 0. We also have: (−1)m ∫ T h B(uh, η)(t)dt+ ∫ QTh uht ηtdxdt = ∫ QTh fηdxdt, for all T > 0, η ∈ Hm,1(eγt, Q∞h ), η(x, t) = 0 with t ≥ T . We have uh(x, t) = 0, fh(x, t) = 0, ∀t ≤ h, and this leads to (−1)m ∫ T −∞ B(uh, η)(t)dt+ ∫ QT−∞ uht ηtdxdt = ∫ QT−∞ fηdxdt, for all T > 0, η ∈ Hm,1(eγt, QR), η(x, t) = 0 with t ≥ T . For f ∈ L2(e−γt, QR), sending h→ −∞, we have (−1)m ∫ T −∞ B(u, η)(t)dt+ ∫ QT−∞ utηtdxdt = ∫ QT−∞ fηdxdt, for all T > 0, η ∈ Hm,1(eγt, QR), η(x, t) = 0 with t ≥ T . This means that u(x, t) is a generalized solution of problem (1.1) - (1.2). We obtain the main result: Theorem 4.1. Take γ > γ0. Assume that (i) sup { ∂apq ∂t , |apq| : (x, t) ∈ QR, 0 ≤ |p|, |q| ≤ m} ≤ µ, (ii) ∂apq ∂t ≤ µ1e2γt, for all (x, t) ∈ QR, 0 ≤ |p|, |q| ≤ m, (iii) f(x, t) ∈ L2(e−γt, QR). Then there exists a unique generalized solution u(x, t) ∈ Hm,1(e−γt, Q∞h ) of problem (1.1) - (1.2) satisfying: ∥u∥2Hm;1(e− t,QR) ≤ C∥f∥2L2(e− t,QR). Acknowledgements. This research was made possible thanks to funding provided by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2011.30. 13 Nguyen Manh Hung and Nguyen Thi Van Anh REFERENCES [1] Adams, R. A., 1975. Sobolev Spaces. Academic Press. [2] A. Kontakov and B. A. Plamenevssky, 2005. 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