tailieunhanh - Báo cáo toán học: "Global Existence of Solution for Semilinear Dissipative Wave Equation"

Trong bài báo này, chúng ta xem xét một vấn đề giá trị ban đầu biên giới cho phương trình sóng semilinear tiêu tán trong một chiều không gian của các loại: utt - uxx + | u | m-1ut = V (t) | u | m-1u + f (t, x) trong (0, ∞) × (a, b) 1 nơi ban đầu dữ liệu u (0, x) = u0 (x) ∈ H0 (a, b), ut (0, x) = u1 (x) ∈ L2 (a, b) và điều kiện biên u (t, a) = u (t, b) = 0 t 0. | Vietnam Journal of Mathematics 34 3 2006 295-305 Viet n a m J 0 u r n a I of MATHEMATICS VAST 2006 Global Existence of Solution for Semilinear Dissipative Wave Equation MD. Abu Naim Sheikh1 and MD. Abdul Matin2 1 Department of Math. Dhaka Univ. of Engineering Technology Gazipur-1700 Bangladesh 2Department of Math. University of Dhaka Dhaka-1000 Bangladesh Received April 29 2005 Abstract. In this paper we consider an initial-boundary value problem for the semilinear dissipative wave equation in one space dimension of the type Utt - uxx u m-1ut V t u m-1u f t x in 0 X X a b where initial data u 0 x Uo x G Ho a b ut 0 x U1 x G L2 a b and boundary condition u t a u t b 0 for t 0 with m 1 on a bounded interval a b . The potential function V t is smooth positive and the source f t x is bounded. We investigate the global existence of solution as t - X under certain assumptions on the functions V t and f t x . 2000 Mathematics Subject Classification 35B40 35L70. Keywords Global existence semilinear dissipative wave equation nonlinear damping potential function source function. 1. Introduction and Results In this paper we consider an initial-boundary value problem for the semilinear dissipative wave equation in one space dimension utt Au Q u ut F u in 0 x X a b u 0 x uo x ut 0 x u1 x for x G a b 1-1 u t a u t b 0 for any t 0 296 MD. Abu Naim Sheikh and MD. Abdul Matin where the function Q u ut u m-1ut represents nonlinear damping and the function F u V t u m- 1u f t x represents source term with m 1 on a bounded interval a b . The potential function V t is smooth positive and f t is a source function which is uniformly bounded as t TO. Georgiev-Todorova 3 treated the case when Q u ut ut m-1ut and F u u p-1u where m 1 and p 1. They proved that if 1 p m a weak solution exists globally in time. On the other hand they also proved that if 1 m p the weak solution blows up in finite time for sufficiently negative initial energy . . 2 . E1 0 IIu1 Il2 11 0 2 0 - p 1 llu0 IL 1 Q . An