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asymptotic stability; viscoelastic problems; boundary dissipation; wave equation
We consider the damped semilinear viscoelastic wave equation \[ u^{\prime \prime } - \Delta u + \int ^t_0 h (t-\tau ) \div \lbrace a \nabla u(\tau ) \rbrace \mathrm{d}\tau + g(u^{\prime }) = 0 \quad \text{in}\hspace{5.0pt}\Omega \times (0,\infty ) \] with nonlocal boundary dissipation. The existence of global solutions is proved by means of the Faedo-Galerkin method and the uniform decay rate of the energy is obtained by following the perturbed energy method provided that the kernel of the memory decays exponentially.
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