Asymptotic behavior of solutions to perturbed superstable wave equations

The paper deals with initial-boundary value problems for the linear wave equation whose solutions stabilize to zero in a finite time. We prove that problems in this class remain exponentially stable in L ² as well as in C ² under small bounded perturbations of the wave operator. To show this for C ², we prove a smoothing result implying that the solutions to the perturbed problems become eventually C ²-smooth for any H ¹ × L ²-initial data.