UNIQUENESS IN A CAUCHY PROBLEM FOR REACTION–DIFFUSION SYSTEM AND INVERSE SOURCE PROBLEMS IN WATER POLLUTION

Abstract

The purpose is a uniqueness result for an ill-posed multi-dimensional parabolic system arising in pollution modeling of surface waters like lakes, estuaries or bays. Exploring organic pollution effects mostly needs the recovery of two tracers, the biochemical oxygen demand (BOD) and the dissolved oxygen (DO) densities when only measurements on the dissolved oxygen concentration are available. The particularity of the reaction–diffusion system resides then in the nature of the boundary conditions. Missing boundary data on BOD is compensated by over-determined boundary conditions on DO which induces a strong coupling in the system. We check first the ill-posedness of the problem. Then, a uniqueness theorem is stated. The saddle point theory and tools from the semi-group analysis turn out to be at the basis of the proof. The results established here may serve for the data completion and the identifiability of pointwise pollution sources in multi-dimensional water bodies.