A uniqueness result for a nonlinear hyperbolic equation

Abstract

In this paper we consider the problem of identifying a coefficient function in the principal part of a nonlinear hyperbolic PDE from overdetermined boundary data of the PDE solution. Assuming that the coefficient function can be written as a linear combination of finitely many ansatz functions, we derive a stability and uniqueness result that is based on an identifiability condition in terms of the initial data as well as on the smallness of the time interval. In doing so, we distinguish between the two- and higher dimensional case where functionals on the boundary can be measured and the one-dimensional situation with only point measurements at one boundary point. Our results also give perspectives to the case of an infinite dimensional coefficient function.