## Precise energy decay rates for some viscoelastic and thermo-viscoelastic rods

##### Abstract

^{-Î±}+ E, where a > 0, E > 0 is examined. In this case, the nonoscillatory modes (the so-called creep modes) dominate the energy decay rate. The results are in two parts.

In the first part, a linear viscoelastic wave equation with infinite memory is examined.
It is shown that under appropriate conditions on the kernel and initial history, the total
energy is integrable against a particular weight if the kinetic energy component of the
total energy is integrable against the same weight. The proof uses energy methods in an
induction argument. Precise energy decay rates have recently been obtained using boundary
velocity feedback. It is shown that the same decay rates hold for history value problems
with conservative boundary conditions provided that an *a priori* knowledge of the decay
rate of the kinetic energy term is assumed.

In the second part, a simple linear thermo-viscoelastic system, namely, a viscoelastic
wave equation coupled to a heat equation, is examined. Using Laplace transform methods,
an integral representation formula for *W(x,s*), the transform of the displacement *w(x, t)*,
is obtained. After analyzing the location of the zeros of the appropriate characteristic
equation, an asymptotic expansion for the displacement *w(O,t)* is obtained which is valid
for large *t* and the specific kernel *g(t) = g*(â ) + Î´tÎ·-1 [over]Î (Î·), 0 < Î· < 1. With this expansion it
is shown that the coupled system tends to its equilibrium at a slower rate than that of the
uncoupled system.

##### Collections

- Doctoral Dissertations [12741]