Energy bounds for some nonstandard problems in partial differential equations

We consider problems of the form d2u dt2 + 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.  2002 Elsevier Science (USA). All rights reserved.