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Vietnam Journal of Mathematics 34:3(2006) 295-305

Global Existence of Solution for Semilinear Dissipative Wave Equation

MD. Abu Naim Sheikh and MD. Abdul Matin

Abstract. In this paper, we consider an initial--boundary value problem for the semilinear dissipative wave equation in one space dimension of the type : $$u_{tt}-u_{xx}+|u|^{m-1}u_t=V(t)|u|^{m-1}u+f(t,x)\quad{\text{in}}\quad(0,\infty) \times(a,b)$$ where initial data $u(0,x) =u_0(x)\in H_0^1(a,b)$, $u_t(0,x)=u_1(x)\in L^2(a,b)$ 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\rightarrow \infty $ 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.