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...
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Báo cáo toán học: "Global Existence of Solution for Semilinear Dissipative Wave Equation" Vietnam Journal of Mathematics 34:3 (2006) 295–305 9LHWQD P-RXUQDO RI 0$7+(0$7, &6 ‹ 9$ 67 Global Existence of Solution for Semilinear Dissipative Wave Equation MD. Abu Naim Sheikh1 and MD. Abdul Matin2 Department of Math., Dhaka Univ. of Engineering & Technology 1 Gazipur-1700, Bangladesh 2 Department of Math., University of Dhaka, Dhaka-1000, Bangladesh Received April 29, 2005Abstract. In this paper, we consider an initial–boundary value problem for thesemilinear 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, ∞) × (a, b)where initial data u(0, x) = u0 (x) ∈ H0 (a, b), ut (0, x) = u1 (x) ∈ L2(a, b) and 1boundary condition u(t, a) = u(t, b) = 0 for t > 0 with m > 1, on a boundedinterval (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 → ∞ under certainassumptions 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 ResultsIn this paper, we consider an initial–boundary value problem for the semilineardissipative wave equation in one space dimension   utt − ∆u + Q(u, ut) = F (u) in (0, ∞) × (a, b),  u(0, x) = u0 (x), ut(0, x) = u1(x) for x ∈ (a, b), (1.1)   u(t, a) = u(t, b) = 0 for any t > 0,296 MD. Abu Naim Sheikh and MD. Abdul Matinwhere the function Q(u, ut) = |u|m−1ut represents nonlinear damping and thefunction F (u) = V (t)|u|m−1u + f (t, x) represents source term with m > 1, ona bounded interval (a, b). The potential function V (t) is smooth, positive andf (t, ·) is a source function, which is uniformly bounded as t → ∞. Georgiev–Todorova [3] treated the case when Q(u, ut) = |ut|m−1 ut andF (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 provedthat if 1 < m < p, the weak solution blows up in finite time for sufficientlynegative initial energy 2 u0 L+1 (Ω) . p E1(0) = u1 2 2 (Ω) + u0 2 2 (Ω) − L L p+1 p+1An extension of Georgiev–Todorova’s blow up result was studied in Levine–Serrin [8], where, among other things, it was shown that if initial energy isnegative, the solution is not global (blow up). Recently, the blow-up result ofGeorgiev–Todorova [3] has been improved also by Sheikh [11]. Ikehata [4] andIkehata–Suzuki [5] considered the case when Q(u, ut) = ut and F (u) = |u|m−1u.They proved that the solution is global and local solution blows up in finite timeby the concepts of stable and unstable sets due to Payne–Sattinger [10]. Lions–Strauss [9] considered the case when Q(u, ut) = k|u|m−1ut and F (u) =f (t, x), where m > 1 and k is a positive constant and proved that a solution existsglobally in time. On the other hand, Katayama–Sheikh–Tarama [6] treated theCauchy and mixed problems in one space dimensional case when Q(u, ut) =k|u|m−1ut and F (u) = k1|u|p−1u, where m > 1, p > 1 and k, k1 are positiveconstants. They proved that if 1 < p m, a weak solution exists globally intime for any initial data. They also proved that if 1 < m < p, the weak solutionblows up in finite time for initial data with bounded support and negative initialenergy 2k1 u0 L+1 (a,b) . p E2(0) = u1 2 2 (a,b) + u0,x 2 2(a,b) − L L p+1 p+1Here we remark that Levine–Serrin [8] considered some evolution equations withQ(u, ut) = |u|κ|ut|m ut and F (u) = |u|p−1u as an example. They proved thatif p > κ + m + 1, the solution is not global for negative initial energy (seealso Levine–Pucci–Serrin [7]). Recently, Georgiev–Milani [2] treated the casewhen Q(u, ut) = |ut|m−1 ut and F (u) = V (t)|u|m−1u + f (t, x). They inves-tigated the asymptotic behavior ...