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

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. | JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci. 2013 Vol. 58 No. 7 pp. 3-14 This paper is available online at http 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 lt 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@ 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

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