TY - JOUR

T1 - Exponential decay of energy of vibrating strings with local viscoelasticity

AU - Liu, K.

AU - Liu, Z.

PY - 2002/3/1

Y1 - 2002/3/1

N2 - In our previous paper, we studied the energy decay rate for an one-dimensional linear wave equation with the Kelvin-Voigt damping presented on a subinterval which models an elastic string with one segment made of viscoelastic material and the other of elastic material. It was proved that the energy of that system does not decay exponentially when each segment is homogeneous, i.e., coefficient functions are piecewise constant and have discontinuity at the interface. This is a puzzling result since it is not seen for the local viscous damping, where the well-known "geometric optics" condition applies. In order to get a better understanding of the causes for this phenomenon, we study two related problems in this paper. We first reconsider the above system with smooth coefficient functions. Then we replace the Kelvin-Voigt model by the Boltzmann model and allow discontinuity of material properties at the interfaces. Exponential energy decay is proved for both cases. These new results suggest that discontinuity of material properties at the interface and the "type" of the damping can affect the qualitative behavior of the energy decay.

AB - In our previous paper, we studied the energy decay rate for an one-dimensional linear wave equation with the Kelvin-Voigt damping presented on a subinterval which models an elastic string with one segment made of viscoelastic material and the other of elastic material. It was proved that the energy of that system does not decay exponentially when each segment is homogeneous, i.e., coefficient functions are piecewise constant and have discontinuity at the interface. This is a puzzling result since it is not seen for the local viscous damping, where the well-known "geometric optics" condition applies. In order to get a better understanding of the causes for this phenomenon, we study two related problems in this paper. We first reconsider the above system with smooth coefficient functions. Then we replace the Kelvin-Voigt model by the Boltzmann model and allow discontinuity of material properties at the interfaces. Exponential energy decay is proved for both cases. These new results suggest that discontinuity of material properties at the interface and the "type" of the damping can affect the qualitative behavior of the energy decay.

KW - Exponential stability

KW - Local viscoelasticity

KW - Semigroup

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U2 - 10.1007/s00033-002-8155-6

DO - 10.1007/s00033-002-8155-6

M3 - Article

AN - SCOPUS:0036005597

VL - 53

SP - 265

EP - 280

JO - Zeitschrift fur Angewandte Mathematik und Physik

JF - Zeitschrift fur Angewandte Mathematik und Physik

SN - 0044-2275

IS - 2

ER -