Abstract

The authors prove the global existence and uniqueness of weak solutions for the linear wave equation with the initial-boundary value problem. The main difficult encountered this paper is the boundary condition at \(x=1\). In order to solve this particular difficulty, they used more stronger assumptions on the initial conditions \(u_0\) and \(u_1\). Moreover, they also studied the regularity of the solution, the stability problem of the solution and the asymptotic expansion of the solution of the linear wave equation with boundary value problem. In this paper, the results are considered as generalization of \textit{N. T. An} and \textit{N. D. Trie} [J. Mech. NCSR, Vietnam Tone. 13(2), 1–7 (1991)] and of \textit{M. Bergounioux}, \textit{N. T. Long} and \textit{A. P. N. Dinh} [Nonlinear Anal., Theory Methods Appl. 43, No. 5 (A), 547–561 (2001; Zbl 0967.35084)], \textit{A. P. N. Dinh} and \textit{N. T. Long} [Demonstr. Math. 19, 45–63 (1986; Zbl 0619.35016)].