Stabilization of a coupled wave equation with one localized nonregular fractional Kelvin–Voigt damping with nonsmooth coefficients

Abstract

In this paper, we study the stabilization of a coupled wave system formed by one localized nonregular fractional viscoelastic damping of Kelvin–Voigt type and localized nonsmooth coefficients. Our main aim is to prove that the C0$$ {C}_0 $$‐semigroup associated with this model is strong stability and decays polynomially at a rate of t−1$$ {t}^{-1} $$. By introducing a new system to deal with fractional Kelvin–Voigt damping, we obtain a new equivalent augmented system, so as to show the well‐posedness of the system based on Lumer–Phillips theorem. We achieve the strong stability for the C0$$ {C}_0 $$‐semigroup associated with this new model by using a general criteria of Arendt–Batty and then turn out a polynomial energy decay rate of order t−1$$ {t}^{-1} $$ with the help of a frequency domain approach.