Polynomial Stabilization of Solutions to a Class of Damped Wave Equations

2017/03/06 by Otared Kavian, Qiong Zhang, Kavian, Otared +1
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.1703.01735

arxiv created 2017/03/06 · arxiv updated 2017/03/07

Abstract

We consider a class of wave equations of the type ∂_tt u + Lu + B∂_t u = 0, with a self-adjoint operator L, and various types of local damping represented by B. By establishing appropriate and raher precise estimates on the resolvent of an associated operator A on the imaginary axis of \Bbb C, we prove polynomial decay of the semigroup exp(-tA) generated by that operator. We point out that the rate of decay depends strongly on the concentration of eigenvalues and that of the eigenfunctions of the operator L. We give several examples of application of our abstract result, showing in particular that for a rectangle Ω:= (0,L₁)× (0,L₂) the decay rate of the energy is different depending on whether the ratio L₁²/L₂² is rational, or irrational but algebraic.

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