Energy bounds for some nonstandard problems in partial differential equations

Abstract

We consider problems of the form d 2u dt 2 +Au=F, αu(0)+u(T)=g, β du dt (0)+ du dt (T)=h, for t∈(0, T), where A is a densely defined, linear, time independent, positive definite symmetric operator and α and β are constants. Although most of our results would hold for more general operators A, we restrict attention to the case in which A is a differential operator and determine ranges of values of α and β for which it is possible to obtain energy bounds, uniqueness results, and, in a special case, pointwise bounds. Some extensions which include a damping term or a term which arises in a generalization of the Kirchhoff string model are also discussed.