Estimations on the exponential decay rate of energy for linear dissipative wave equations. (Estimations sur le taux de décroissance exponentielle de l’énergie dans des équations des ondes dissipatives linéaires.) (French. Abridged English version) Zbl 0796.35098

Cox, Steven; Zuazua, Enrique

C. R. Acad. Sci., Paris, Sér. I 317, No. 3, 249-254 (1993).

Summary: We consider the linear dissipative wave equation in a bounded domain of \(\mathbb{R}^ n\) with Dirichlet boundary condition. The exponential decay rate of the energy is studied. In one space dimension \((n=1)\) and when the damping is of bounded variation, we prove that the decay rate coincides with the spectral abscissa. We also obtain the asymptotic form of eigenvalues and eigenfunctions corresponding to high frequencies. As an immediate consequence of these asymptotics we show that the decay rate is bounded above by the total amount of damping. Finally, in any space dimension, we discuss low frequencies and conditions for over- or under- damping.

MSC:

35L20	Initial-boundary value problems for second-order hyperbolic equations
35P20	Asymptotic distributions of eigenvalues in context of PDEs

Keywords:

exponential decay rate of the energy; asymptotic form of eigenvalues and eigenfunctions