Mathematical Control and Related Fields (MCRF)

Energy decay estimates for some weakly coupled Euler-Bernoulli and wave equations with indirect damping mechanisms

Pages: 45 - 60, Volume 2, Issue 1, March 2012      doi:10.3934/mcrf.2012.2.45

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Louis Tebou - Department of Mathematics & Statistics, Florida International University, Miami, FL 33199, United States (email)

Abstract: We consider the Euler-Bernoulli equation coupled with a wave equation in a bounded domain. The Euler-Bernoulli has clamped boundary conditions and the wave equation has Dirichlet boundary conditions. The damping which is distributed everywhere in the domain under consideration acts through one of the equations only; its effect is transmitted to the other equation through the coupling. First we consider the case where the dissipation acts through the Euler-Bernoulli equation. We show that in this case the coupled system is not exponentially stable. Next, using a frequency domain approach combined with the multiplier techniques, and a recent result of Borichev and Tomilov on polynomial decay characterization of bounded semigroups, we provide precise decay estimates showing that the energy of this coupled system decays polynomially as the time variable goes to infinity. Second, we discuss the case where the damping acts through the wave equation. Proceeding as in the first case, we prove that this new system is not exponentially stable, and we provide precise polynomial decay estimates for its energy. The results obtained complement those existing in the literature involving the hinged Euler-Bernoulli equation.

Keywords:  Euler-Bernoulli equation, plate equation, wave equation, polynomial decay estimates, coupled equations, indirect damping, frequency domain method, multiplier techniques.
Mathematics Subject Classification:  Primary: 93D15; Secondary: 35L05, 35Q74, 37L15, 74K15, 74K20.

Received: August 2011;      Revised: December 2011;      Available Online: January 2012.