SHARPER AND FINER ENERGY DECAY RATE FOR AN ELASTIC STRING WITH LOCALIZED KELVIN-VOIGT DAMPING

This paper is on the asymptotic behavior of the elastic string equation with localized Kelvin-Voigt damping utt(x, t) − [ux(x, t) + b(x)ux,t(x, t)]x = 0, x ∈ (−1, 1), t > 0, where b(x) = 0 on x ∈ (−1, 0], and b(x) = a(x) > 0 on x ∈ (0, 1). It is known that the Geometric Optics Condition for exponential stability does not apply to Kelvin-Voigt damping. Under the assumption that a⁰(x) has a singularity at x = 0, we investigate the decay rate of the solution which depends on the order of the singularity. When a(x) behaves like x^α(− log x)^−β near x = 0 for 0 ≤ α < 1, 0 ≤ β or 0 < α < 1, β < 0, we show that the system can achieve a mixed polynomial-logarithmic decay rate. 3−α−ε As a byproduct, when β = 0, we obtain the decay rate t⁻²⁽¹−α) of solution 1 for arbitrarily small ε > 0, which improves the rate t⁻¹⁻α obtained in [14]. The new rate is again consistent with the exponential decay rate in the limit case α → 1⁻. This is a step toward the goal of obtaining the optimal decay rate eventually.