On the Linear Damped Wave Equation

In this work we estimate the spectrum of the linear damped wave semigroup under homogeneous Dirichlet boundary conditions by using the principal eigenvalue of an elliptic operator related to the equation. Our estimate is optimal for real eigenvalues. Then, we analyze the behavior of the estimate as thedamping amplitudegrows to infinity. When the damping changes of sign we extend a result of Freitas [5] to show that the semigroup possesses at least two real eigenvalues greater than one if the amplitude is sufficiently large. In particular, the trivial state is unstable, in strong contrast with the sign definited case. Finally, we characterize the limiting behavior of the real eigenpairs which originate the inestability of the trivial state. This analysis is based upon the behavior of the principal eigenpair of a singular perturbation problem at the singular limit. Our theory is of interest by itself and it has many applications to reaction diffusion systems (c.f. for instance [6]).