ABSTRACT
Determining the asymptotic behaviour of solutions to linear partial differential equations (PDEs) is often a delicate matter. The principle of linear stability holds whenever one has a suitable spectral mapping theorem for the semigroup. This is the case for a wide variety of semigroups. This chapter tries to find conditions on the perturbation guaranteeing equality of the bounds. The well-posedness of this kind of problem is treated in the monograph by Goldstein. The chapter shows that for a class of self-adjoint perturbations the equality of bounds which exists for the wave equation is preserved. It also shows that Renardy’s construction of a counterexample can be extended to higher order equations. It makes use of the theory of cosine functions which was partly developed by Goldstein. The chapter constructs an operator for a fourth order differential equation where the growth bound of the generated semigroup and the spectral bound of the generator differ.
