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We are concerned with the inverse problem of detecting  sources  in a coupled diffusion-reaction system. This problem arises from the  Biochemical Oxygen Demand-Dissolved Oxygen model($^1$) governing the interaction between organic pollutants and the oxygen available in stream waters. The sources we consider are point-wise and  simulate stationary or moving pollution sources. The ultimate objective is  to obtain their discharge location and recover their output rate from accessible measurements of DO when BOD measurements are difficult and time consuming to obtain.  It is, as a matter of fact, the most realistic configuration. The subject to address here is the identifiability of these sources, in other words to determine if the observations uniquely determine the sources. The key tool is the study of coupled parabolic systems derived after restricting the global  model to regions at the exterior of the observations.  The absence of any prescribed  condition on the BOD density is compensated  by data recorded on the DO which provide  over-determined Cauchy boundary conditions. Now, the first  step toward the identifiability of the sources  is precisely to recover the BOD at the observation points (of DO). This may be achieved by handling and solving  the coupled systems. Unsurprisingly, they turn out to be ill-posed. That issue is investigated first. Then, we state  a uniqueness result owing to a suitable saddle-point variational  framework and  to Pazy's uniqueness Theorem. This uniqueness complemented by  former identifiability results proved in [2011, Inverse problems] for scalar  reaction-diffusion equations  yields the desired  identifiability for the global model.

Uniqueness for an ill-posed reaction-dispersion model. Application to organic pollution in stream-waters