tailieunhanh - The first initial boundary value problem for semilinear hyperbolic equations in nonsmooth domains

In this paper we study the first initial boundary problem for semilinear hyperbolic equations in nonsmooth cylinders, where is a nonsmooth domain in Rn, n >=2. We established the existence and uniqueness of a global solution in time. | JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci. 2013 Vol. 58 No. 7 pp. 39-49 This paper is available online at http THE FIRST INITIAL-BOUNDARY VALUE PROBLEM FOR SEMILINEAR HYPERBOLIC EQUATIONS IN NONSMOOTH DOMAINS Vu Trong Luong and Nguyen Thanh Tung Faculty of Mathematics Tay Bac University Abstract. In this paper we study the first initial boundary problem for semilinear hyperbolic equations in nonsmooth cylinders Q Ω 0 where Ω is a nonsmooth domain in Rn n 2. We established the existence and uniqueness of a global solution in time. Keywords Initial boundary value problem semilinear hyperbolic equation global solution non-smooth domain. 1. Introduction Let Ω Rn be a bounded domain with non-smooth boundary Ω set ΩT Ω 0 T with 0 lt T lt . We use the notations H 1 Ω H01 Ω as the usual Sobolev spaces and H 1 Ω as the dual space of H01 Ω is defined in 1 . We denote L2 Ω as the space L2 Ω is defined in 2 . Suppose X is a Banach space with the norm X . The space Lp 0 X consists of all measurable functions u 0 X with norm p1 u Lp 0 T X u t pX dt lt for 1 p lt . 0 We consider the partial differential operator n n u u Lu ij a x t bi x t c x t u i j 1 x j x i i 1 x i where x t Q Ω 0 aij bi c C 1 Q i j 1 n Received March 12 2013. Accepted June 5 2013 Contact Nguyen Thanh Tung e-mail address thanhtung70tbu@ 39 Vu Trong Luong and Nguyen Thanh Tung aij x t aji x t for i j 1 2 n x t Q. The operator L is strongly elliptic. Then there exists θ gt 0 ξ Rn x t Q such that n aij x t ξi ξj θ ξ 2 . i j 1 In this paper we consider the initial-boundary value problem in the cylinder Q for semilinear PDE s utt Lu f x t u Du h x t x t Q u x 0 u0 x ut x 0 u1 x x Ω u x t 0 x t Ω 0 where u0 H01 Ω u1 L2 Ω h L2 0 L2 Ω and f Q R Rn R is continuous and satisfies the following two conditions f x t u Du C k x t u Du x t Q k L2 0 L2 Ω f x t u Du f x t v Dv u v dx 0 . t 0 . Ω We introduce the Sobolev space H 1 1 Q which consists of

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