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

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In this paper, we study the Neumann boundary value problems without initial condition for Hyperbolic systems in cylinders. The main obtained results are the uniqueness and the existence of generalized solutions.

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

JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci., 2013, Vol. 58, No. 7, pp. 3-14 This paper is available online at http://stdb.hnue.edu.vn 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 1 National Institute of Education Management 2 Hanoi 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), Sab = 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 H k,l (QR ) the space consisting of all vector functions u : QR −→ Cs satisfying: ∫ (∑ k ∑ l ) ∥u∥2H k,l (QR ) := |Dα u|2 + |utj |2 dxdt, QR |α=0| j=1 and H k,l (e−γt , QR ) is the space of vector functions with norm ∫ (∑ k ∑ l ) ∥u∥2H k,l (e−γt ,QR ) := |D u| + α 2 |utj | e−2γt dxdt. 2 QR |α=0| j=1 In particular ∫ ∑ k ∥u∥2H k,0 (e−γt ,QR ) := |Dα u|2 e−2γt dxdt. QR |α=0| Especially, we set L2 (e−γt , QR ) = H 0,0 (e−γt , QR ). We denote by { } Hk,l (e−γt , QR ) = u ∈ H k,l (e−γt , QR ) such that lim ∥u(., t)∥L2 (Ω) = 0 , t→−∞ so Hk,l (e−γt , QR ) is a linear space. Adding that, we consider an important space H k,l (eγt , QR ) to be the space of vector functions with norm ∫ (∑ k ∑ l ) ∥u∥2H k,l (eγt ,QR ) := |D u| + α 2 |utj |2 e2γt dxdt < +∞. QR |α=0| j=1 In particular ∫ ∑ k ∥u∥2H k,0 (eγt ,QR ) := |Dα u|2 e2γt dxdt < +∞. QR |α|=0 We introduce the matrix differential operator: ∑ m L(x, t, D) = Dp (apq (x, t)Dq ), |p|,|q|=0 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 ∈ H 2m (Ω), we have: ∫ ∑ m ∫ ( ∑ m ) aαβ (x)D β uDα vdx = (−1)|α| Dα (aαβ (x)Dβ u) vdx Ω |α|,|β|=0 Ω |α|,|β|=0 ∫ ∑ m − (Bj v)(Nj u)dS. j=1 ∂Ω 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 ∑ m ∫ |p| B(u, v)(t) = (−1) apq Dq uDp vdx, |p|,|q|=0 Ω and { } HNm (Ω) = u ∈ H m (Ω) : Nj u = 0 on S for all j = 1, . . . , m . We assume further that the form (−1)m B(., .)(t) is HNm - uniformly elliptic with respect to t and that means there exists a constant µ0 > 0 independent of t and u such that: (−1)m B(u, u)(t) ≥ µ0 ∥u(., t)∥2H m (Ω) for all u ∈ HNm (Ω), and a.e t ≥ h. We consider the hyperbolic system in the cylinder QR (−1)m−1 L(x, t, D)u − utt = f (x, t) in QR , (1.1)
Nj u
= 0, j = 1, . . . , m. (1.2) SR 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: ∫ T ∫ ∫ m−1 (−1) B(u, η)(t)dt + ut ηt dxdt = f ηdxdt, (1.3) −∞ QT −∞ QT −∞ holds for all η ∈ H m,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)
∂apq
Theorem 2.1. If γ > 0 and
< µ1 e2γt , ∀t ∈ R, ∀|p|, |q| ≤ m, the problem ∂t (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),  ∫ u(x, τ )dτ,  t −∞ ≤ t ≤ b, η(x, t) = b  0 , b ≤ t ≤ T. So we get η(x, T ) = 0, η(x, t) ∈ H m,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: ∑m ∫ ∫ m−1 |p| q p (−1) (−1) apq D ηt D ηdxdt + ηtt η t dxdt = 0. (2.1) |p|,|q|=0 Qb−∞ Qb−∞ Adding the equation (2.1) with its complex conjugate we get: ∫ b ∫ m−1 (−1) 2Re B(ηt , η)(t)dt + 2Re ηtt η t dxdt = 0. (2.2) −∞ Qb−∞ 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: ∑ m ∫ |p| ∂apq q p ∥ηt (., b)∥L2 (Ω) + lim (−1) B(η, η)(h) = (−1) 2 m m−1 (−1) D ηD ηdxdt. h→−∞ Qb−∞ ∂t |p|,|q|=0 Noting the asumption, we then have the fact that the coeficients apq are continuous with respect to the time variable and η ∈ H m,1 (eγt , QT−∞ ), so there exists the limit lim (−1)m B(η, η)(h). By using a uniformly elliptic condition we imply: h→−∞ lim (−1)m B(η, η)(h) ≥ µ0 lim ∥η(., h)∥2H m (Ω) , h→−∞ h→−∞ and thus ∑ m ∫ |p| ∂apq q p ∥ηt (., b)∥2L2 (Ω) +µ0 lim ∥η(., h)∥2H m (Ω) ≤ (−1) m−1 (−1) D ηD ηdxdt. h→−∞ Qb−∞ ∂t |p|,|q|=0 6
On the solvability of the neumann boundary value problem without initial conditions... By using the Cauchy inequality, we have:
∑m ∫
∫ b
|p| ∂apq q p
∗
(−1) m−1 (−1) D ηD ηdxdt
≤ µ1 m ∥η(., t)∥2H m (Ω) e2γt dt. |p|,|q|=0 Qb−∞ ∂t −∞ We have yields: ∫ b ∗ ∥ηt (., b)∥2L2 (Ω) + µ0 lim ∥η(., h)∥2H m (Ω) ≤ µ1 m ∥η(., t)∥2H m (Ω) e2γt dt. (2.3) h→−∞ −∞ Now denote by: ∫ h vp (x, t) = Dp u(x, τ )dτ, −∞ ≤ t ≤ b. t Hence, we can see that ∫ t p D η(x, t) = Dp u(x, τ )dτ = vp (x, b) − vp (x, t), lim Dp η(x, h) = vp (x, b), b h→−∞ m ∫ ∑ lim ∥η(., h)∥2H m (Ω) = |vp (x, b)|2 dx, h→−∞ |p|=0 Ω and from the equality (2.3) we have: m ∫ ∑ ∫ b ∗ ∥ηt (., b)∥L2 (Ω) + µ0 2 |vp (x, b)| dx ≤ µ1 m 2 ∥η(., t)∥2H m (Ω) e2γt dt. |p|=0 Ω −∞ This then leads to ∑ m m ∫ ∑ ∗ ∥ηt (., b)∥2L2 (Ω) + µ0 ∥vp (x, b)∥2L2 (Ω) ≤ µ1 m e2γt |Dp η(x, t)|2 dxdt |p|=0 |p|=0 Qb−∞ ∑ m m ∫ ∑ b ∗ 2γb ∗ ≤ 2µ1 m e ∥vp (., b)∥2L2 (Ω) + 2µ1 m e2γt ∥vp (., t)∥2L2 (Ω) dt |p|=0 |p|=0 −∞ ∑ m =⇒∥ηt (., b)∥2L2 (Ω) + (µ0 − 2µ1 m∗ e2γb ) ∥vp (x, b)∥2L2 (Ω) |p|=0 ∫ ∑ 2µ1 m ∗ b ( ∗ 2γb m ) ≤ e 2γt ∥ηt (., t)∥2L2 (Ω) + (µ0 − 2µ1 m e 2γt )e ∥vp (x, b)∥2L2 (Ω) dt. µ0 − 2µ1 m∗ e2γb −∞ |p|=0 µ0 So, there exists a positive number C > 0, C > such that 2µ1 m∗ ∑ ∫ ∑ m b ( m ) ∥ηt (., b)∥2L2 (Ω) + ∥vp (x, b)∥2L2 (Ω) ≤C e 2γt ∥ηt (., t)∥2L2 (Ω) + ∥vp (x, b)∥2L2 (Ω) dt. |p|=0 −∞ |p|=0 7
Nguyen Manh Hung and Nguyen Thi Van Anh Put ∑ m J(t) = ∥ηt (x, t)∥2L2 (Ω) + (µ0 − 2µ1 m∗ e2γb ) ∥vp (x, t)∥2L2 (Ω) , |p|=0 we have: ∫ b 1 1 J(b) ≤ C e2γt J(t)dt, for a.e b ≤ ln . −∞ 2γ C By performing a check similar to the proof of the Gronwall- Bellman inequality (see [4], page 624-625), we will prove that 1 1 J(t) ≡ 0 on (−∞, ln ]. 2γ C ∫t ∫0 In fact, taking ζ(t) = −∞ e2γs J(s)ds, we have ζ ′ (t) = ( e2γs J(s)ds + ∫ t 2γs −∞ 0 e J(s)ds)′ = e2γt J(t), then we have: 1 1 ζ ′ (t) ≤ Ce2γt ζ(t) for a.e t ≤ ln . 2γ C From this we see d( −Ce2γs ) −Ce2γs ζ(s)e 2γ = e 2γ (ζ ′ (s) − Ce2γt ζ(t)) ≤ 0. ds −Ce2γs By integrating with respect to s from −∞ to t in remark that lim ζ(s)e 2γ = 0, we s→−∞ get −Ce2γt 1 1 ζ(t)e 2γ ≤ 0 for a.e t ≤ ln . 2γ C Thus we obtain ζ(t) ≤ 0 and we can conclude ζ ′ (t) ≤ 0 for a.e t ≤ 2γ 1 ln C1 by the above estimate. From this, one has the desied estimate. 1 1 So u(x, t) = 0 almost everywhere t ∈ (−∞, ln ]. Because of the uniqueness of the 2γ C 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)m Lu − utt − αut = f, (x, t) ∈ QR , (i)
Nj u