Asymptotics of Perturbations to the Wave Equation

The starting point for this article is a well-known example by M.~Renardy showing the failure of the equality $\omega(T)=s(A)$ for a first order perturbation to the wave equation, where $\omega(T)$ denotes the growth bound of the semigroup $T$ generated by $A$ and $s(A)$ is the spectral bound of $A$. In this article we give conditions on first order perturbations to the wave equation guaranteeing the equality. More specifically, we show that for a class of self-adjoint perturbations the equality of bounds which exists for the wave equation is preserved. Making use of the theory of cosine functions, we are able to extend Renardy's construction of a counterexample to higher order equations.