Unicity in an inverse problem for an unknown reaction term in a reaction-diffusion equation

Abstract

In this paper we consider an inverse problem in which we seek to deter- mine an unknown source or reaction term in a reaction-diffusion equation from overspecified data measured on the boundary of the spatial region where the equation applies. The analysis is based on the observation that the overspecified data depends monotonically on the unknown source term in the equation. Here we use this monotonicity to establish a type of unicity result for the inverse problem. The attractiveness of the monotonicity methods illustrated here lies in their versatility. Surveying the rapidly expanding literature devoted to the topic of inverse problems, one finds a considerable variety of ad hoc approaches. Methods which apply to a general class of problems seem to be few. Monotonicity methods, however, apply to a variety of inverse problems involving partial differential equations of parabolic or elliptic type. This paper is organized as follows. In Section 1 we formulate the direct initial boundary value problem (IBVP) and state assumptions on the data and a priori assumptions on the unknown source term. Several properties of the solution to the direct problem are derived. In Section 2 the principal monotonicity estimates are established and in Section 3 these are applied to the inverse problem to prove a type of unicity result for the solution to the inverse problem. In particular, it is a Corollary of Theorem 3.2 that in the class of analytic source terms, the solution of the inverse problem is unique. The problem of identifying an unknown source term in a heat equation 155