Perturbing the critically damped wave equation

We consider the wave equation with viscous damping. The equation is said to be critically damped when the damping is that value for which the spectral abscissa of the associated wave operator is minimized within the class of constant dampings. The critically damped wave operator possesses a nonsemisimple eigenvalue. We present a detailed study of the splitting of this eigenvalue under bounded perturbations of the damping and subsequently show that the critical choice is a local minimizer of the spectral abscissa over lines in the class of all bounded dampmgs.