Stabilization of Viscoelastic Wave Equations with Distributed or Boundary Delay

Abstract

The wave equation with viscoelastic boundary damping and internal or boundary delay is considered. The memory kernel is assumed to be integrable and completely monotonic. Under certain conditions on the damping factor, delay factor and the memory kernel it is shown that the energy of the solutions decay to zero either asymptotically or exponentially. In the case of internal delay, the result is obtained through spectral analysis and the Gearhart-Prüss Theorem, whereas in the case of boundary delay, it is obtained using the energy method.

Asymptotic stability of the multidimensional wave equation coupled with classes of positive-real impedance boundary conditions

“…Let us define u ∈ ∇H 1 (Ω) by (50a). Taking ψ ∈ C ∞ 0 (Ω) in (53) shows that u ∈ H div (Ω) and (50b) holds. Using the expressions of u and div u, and Green's formula (72), the weak formulation (53) can be rewritten as…”

“…Integrable kernels coupled with a 2-dimensional realization are considered in [35] using energy estimates. Kernels that are both completely monotone and integrable are considered in [16], which uses the ABLV theorem on an extended state space, and in [53] with an added time delay, which uses the energy method to prove exponential stability. The energy multiplier method is also used in [4] to prove exponential stability for a class of non-integrable singular kernels.…”

“…This result covers both distributed and discrete time delays, as well as a class of integrable kernels. Other classes of integrable kernels have been studied in [16,53,35]. Integrable kernels coupled with a 2-dimensional realization are considered in [35] using energy estimates.…”