Let be a bounded (smooth) domain in and let I be either interval (- or (0,T) . The author considers the following identification problems determine a pair of functions u: and f: such that
Here , is a uniformly elliptic operator with negative definite principal part, while and with for some positive constant , N(x) denoting the outward normal at a point (x,t). Finally, are prescribed functions in suitable (anisotropic) Hölder spaces.
Under suitable hypotheses on A,B and , the author shows that problem (1)-(4) is Fredholm with zero index. Then, by restriction of the classes of admissible unknowns f and weights , the author proves the uniqueness (and, consequently, the existence and the stability) of the solution to problem (1)-(4). Two basic assumptions for this are the positivity and the strict monotonicity of with respect to time , . In connection with such hypotheses, the author shows that, when fails to be strictly positive in there exist positive -weight functions for which problem (1)-(4) admits nontrivial solutions.
A further identification problem is studied in the case where and the coefficient a in A is supposed to be unknown and to depend only on x. Also for such a problem existence and uniqueness results are proved.
The paper concludes with the proof of a stability result for the solution to problem (1)-(4), when Q is a non-cylindrical domain.