Local uniform stability for the semilinear wave equation in inhomogeneous media with locally distributed Kelvin–Voigt damping

Abstract

We consider the semilinear wave equation posed in an inhomogeneous medium Ω with smooth boundary ∂Ω subject to a local viscoelastic damping distributed around a neighborhood ω of the boundary according to the Geometric Control Condition. We show that the energy of the wave equation goes uniformly and exponentially to zero for all initial data of finite energy taken in bounded sets of finite energy phase‐space. As far as we know, this is the first stabilization result for a semilinear wave equation with localized Kelvin–Voigt damping.

Exponential decay for semilinear wave equation with localized damping in the hyperbolic space

“…The second one is an assumption, which involves a unique continuation result for an equation with nonhomogeneous second-order operator. See also Astudillo et al [1], Cavalcanti et al [13], and references therein. Finally, we cite the work of Yao [37], which introduced the geometric multipliers to deal with boundary exact controllability for wave equations with nonconstant coefficients in the principal part.…”

Exponential decay for the semilinear wave equation with localized Kelvin-Voight damping

“…More recently, Astudillo et al [3] proved the exponential stability when f (s) = 0, in an inhomogeneous medium, for a particular class of density functions ρ(x), which helped to control the bicharacteristic flow by controlling its projection on the spatial domain, for dimensions n ≥ 2.…”