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Long time behavior for a wave equation with time delay. (English) Zbl 1361.35105

Summary: In this paper, we consider the wave equation with internal time delay and source terms \( u_{tt}(x,t) - u(x,t) + \mu_1 u_t(x,t) + \mu_2 u_t(x,t-\tau) + f(x,u) = h(x)\) in a bounded domain. By virtue of Galerkin method combined with the priori estimates, we prove the existence and uniqueness of global solution under initial-boundary data for the above equation. Moreover, under suitable conditions on the forcing term \(f(x,u)\) and \(\mu_1\), \(\mu_2\), the existence of a compact global attractor is proved. Further, the asymptotic behavior and the decay property of global solution are discussed.

MSC:

35L20 Initial-boundary value problems for second-order hyperbolic equations
35R10 Partial functional-differential equations
35B41 Attractors