Stability of a String with Local Kelvin--Voigt Damping and Nonsmooth Coefficient at Interface

Optimal polynomial stability of a string with locally distributed Kelvin–Voigt damping and nonsmooth coefficient at the interface

“…This result is due to the discontinuity of the viscoelastic materials. Later, in the one-dimensional case, it was found that the smoothness of the damping coefficient at the interface is an essential factor for the stability and the regularity of the solutions (see previous studies [2][3][4][5][6][7] ). Let us begin by recalling previous studies done on one-dimensional wave equation with Kelvin-Voigt damping.…”

Logarithmic Decay of Wave Equation with Kelvin-Voigt Damping

“…By Gårding inequality (11), there exists a constant C > 0 such that, for τ ≥ τ 0 where τ 0 sufficiently large,…”

“…More precisely, assume that Ω = (−1, 1) and a(x) behaviours like x α with α > 0 in supp a = [0, 1]. Then the solution of (1) is eventually differentiable for α > 1, exponentially stable for α ≥ 1, polynomially stable of order 1 1−α for 0 < α < 1, and polynomially stable of optimal decay rate 2 for α = 0 (see [10][11][12][13]).…”

Stability of the wave equations on a tree with local Kelvin–Voigt damping

“…Then, eᾱ may be parametrized by its arc length by means of the functions πᾱ, defined in [0, ᾱ ] such that πᾱ( ᾱ ) = Oᾱ and πᾱ(0) is the other vertex of this edge. Now, we are ready to introduce a planar tree-shaped network of N elastic strings, where N ≥ 2, see [20,22,26,17,19] and [16] concerning the model. More precisely, we consider the following initial and boundary value problem : (1.2) u(0, t) = 0, uᾱ( ᾱ , t) = 0,ᾱ ∈ I S , t > 0, (1.3) uᾱ •β (0, t) = uᾱ( ᾱ , t), t > 0, β = 1, 2, ..., mᾱ,ᾱ ∈ I M ,…”