Longtime behavior of multidimensional wave equation with local Kelvin–Voigt damping

Abstract

In this paper, the longtime behavior of a coupled multidimensional elasticviscoelastic waves system is considered. This model consists of an elastic wave domain and an viscoelastic wave domain, connecting by a common interface.The dissipative damping is produced in the viscoelastic wave via the boundary connection. By the resolvent estimate together with microlocal analysis argument, we show that the corresponding semigroup is polynomially stable with decay rate 𝑡 −1 under certain conditions.

Energy decay for a coupled wave system with one local Kelvin–Voigt damping

“…In [11], Burq showed the following conclusions: if $$\Omega$$ is a cube or $$\Omega _d$$ satisfies the above geometric control condition and $$d$$ satisfies (4), then the energy of the wave equation decays polynomially; if $$d$$ only satisfies (4), then the energy of the wave equation decays logarithmically. For other results of the stability of the wave equation with Kelvin–Voigt damping, we refer the reader to [7, 22, 27, 28, 30, 31, 38]) for exponential stability and [4, 5, 7, 15, 17–19, 29, 31, 32, 37, 39] for polynomially stability, respectively.…”