Stability results of a singular local interaction elastic/viscoelastic coupled wave equations with time delay

Abstract

The purpose of this paper is to investigate the stabilization of a one-dimensional coupled wave equations with non smooth localized viscoelastic damping of Kelvin-Voigt type and localized time delay. Using a general criteria of Arendt-Batty, we show the strong stability of our system in the absence of the compactness of the resolvent. Finally, using frequency domain approach combining with a multiplier method, we prove a polynomial energy decay rate of order t −1 .

On the stability of Bresse system with one discontinuous local internal Kelvin-Voigt damping on the axial force

“…It is easy to see that B is a sesquilinear and continuous form on H 1 0 (0, L) 3 × H 1 0 (0, L) 3 and L is a linear and continuous form on H 1 0 (0, L) 3 . In fact, from Remark 2.1, we deduce that there exists a positive constant…”

“…Thus, B is a coercive form on H 1 0 (0, L) 3 × H 1 0 (0, L) 3 . Then, it follows by Lax-Milgram theorem that (2.18) admits a unique solution (v 1 , v 3 , v 5 ) ∈ H 1 0 (0, L) 3 .…”

“…Thus, B is a coercive form on H 1 0 (0, L) 3 × H 1 0 (0, L) 3 . Then, it follows by Lax-Milgram theorem that (2.18) admits a unique solution (v 1 , v 3 , v 5 ) ∈ H 1 0 (0, L) 3 . By taking test-functions (φ 1 , φ 2 , φ 3 ) ∈ (D(0, L)) 3 , we see that (2.15)-(2.17) hold in the distributional sense, from which we deduce that (v…”

“…In the recent years, many researchers showed interest in problems involving Kelvin-Voigt damping where different types of stability, depending on the smoothness of the damping coefficients, has been showed (see [7], [8], [25], [26], [28], [31], [34], [35], [39] and [41]). Moreover, there is a number of new results concerning systems with local Kelvin-Voigt damping and non-smooth coefficients at the interface (see [3], [5], [20], [21], [22], [27] and [37]).…”

“…3 ), consequently, from Lemmas 4.3, 4.5 with ℓ = 4 and the fact that v 3x , (v 1 x + v 3 + lv 5 are uniformly bounded in L 2 (0, L) and v 3 = O(|λ| −1 ), we obtain(4.56) ℜ − β−3ε α+3ε f 4 v 3 x v 1 xx dx = o(1).…”