Precise energy decay rates for some viscoelastic and thermo-viscoelastic rods
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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.
- Doctoral Dissertations