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The neumann problem for second order hyperbolic equations in nonsmooth domains

In this paper, we study the Neumann problem for second order hyperbolic equations without initial data in nonsmooth domains. Our intension was to prove the existence of a generalized solution to this problem by applying the results of this problem to the initial data. | JOURNAL OF SCIENCE OF HNUE Natural Sci. 2012 Vol. 57 No. 3 pp. 53-59 THE NEUMANN PROBLEM FOR SECOND ORDER HYPERBOLIC EQUATIONS IN NONSMOOTH DOMAINS Nguyen Van Trung Hong Duc University Thanh Hoa City E-mail trungnv2000@gmail.com Abstract. In this paper we study the Neumann problem for second order hyperbolic equations without initial data in nonsmooth domains. Our inten- sion was to prove the existence of a generalized solution to this problem by applying the results of this problem to the initial data. Keywords Neumann problem nonsmooth domains hyperbolic equations generalized solution. 1. Introduction We are concerned with boundary value problems and value for hyperbolic equations in nonsmooth domains. The problems with the Dirichlet boundary condi- tions and initial data have been investigated in 2 3 4 . The boundary value problem without initial condition for parabolic equation has been investigated in 5 6 7 8 . The main goal of this work is to prove the existence of a generalized solution of the Neumann problem without having initial data. Let Ω be a nonsmooth domain in Rn n 2 . For h R set Qh Ω h Sh Ω h S Ω R Q Ω R. Let n X n X L x t i aij x t j bi x t i c x t i j 1 i 1 be a second order partial differential operator where i xi and aij bi c are bounded functions from C Q . We study the present paper hyperbolic equation utt L x t D u f in Q 1.1 with Neumann boundary conditions u x t 0 x t S 1.2 νL where u x t Pn νL j 1 j u x t aij x t νi x t . 53 Nguyen Van Trung Let m k be non-negative integers. Denote by H m Ω the Sobolev spaces in 1 . By the notation . . we mean the inner product in L2 Ω . Denote by H 1 Ω the dual space of H 1 Ω . The pairing between H 1 Ω and H 1 Ω is denoted by h. .i. Let X be a Banach space γ γ t be a real functions. We denote by L2 a γ X the space of functions f a X with the norm Z 21 kf kLp a γ X kf t k2X e γ t t dt lt . a Finally we introduce the Sobolev space H 1 1 Qa γ which consists all functions u defined on Qa such that u L2 a γ