The rate at which energy decays in a string damped at one end. (English) Zbl 0847.35078

From the authors’ abstract: The energy in a nonhomogeneous string that moves freely at one end and against unit friction at the other is known to decay exponentially in time. The energy in fact vanishes in finite time when the string’s density is uniformly one. We show that this is the only distribution of mass for which the energy vanishes in finite time. Assuming that the density is not one at the damped end and that it has two square integrable derivatives we identify the best rate of decay with the supremum of the real part of the spectrum of the infinitesimal generator of the underlying semigroup. Careful estimation of this spectrum permits us to establish the existence of a density that minimizes the decay rate over a large class of competitors.
Reviewer: R.Racke (Konstanz)


35L20 Initial-boundary value problems for second-order hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
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