Abstract
This paper is devoted to the study of a wave equation with a boundary condition of many-point type. The existence of weak solutions is proved by using the Galerkin method. Also, the uniqueness and the stability of solutions are established.
Highlights
Initial-boundary value problems of wave equations have appeared more and more in mechanics, they have been deeply studied by many authors, and we can refer to the works [1,2,3,4,5,6,7,8,9,10,11,12,13]
Nguyen and Giang Vo [8] obtained the asymptotic behavior of the weak solution of the following initialboundary value problem as ε → 0+: utt − uxx + Ku + λut = f (x, t), u (1, t) = 0, ux (0, t) = g (t) + hu (0, t) + εut (0, t)
Here we prove that this solution is stable in the sense of continuous dependence on the given data (f, g, h, k, σ)
Summary
Initial-boundary value problems of wave equations have appeared more and more in mechanics, they have been deeply studied by many authors, and we can refer to the works [1,2,3,4,5,6,7,8,9,10,11,12,13]. U (x, 0) = u0 (x) , ut (x, 0) = u1 (x) , in which g, ρ, u0, u1 are given functions He studied the asymptotic behavior of the solutions of problem (3) with. Takaci [12] gave the formula for finding the exact solutions of the linear wave equation given by utt − uxx + Ku + λut = f (x, t) , ux (0, t) = V (t) ,. Nguyen and Giang Vo [8] obtained the asymptotic behavior of the weak solution of the following initialboundary value problem as ε → 0+: utt − uxx + Ku + λut = f (x, t) , u (1, t) = 0, ux (0, t) = g (t) + hu (0, t) + εut (0, t). This paper is a relative generalization of the works [4, 6, 11, 12]
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