# American Institute of Mathematical Sciences

May  2012, 6(2): 163-181. doi: 10.3934/ipi.2012.6.163

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

 1 LMAC, EA 2222, Université de Technologie de Compiègne, BP 20529, 60205 COMPIEGNE Cedex, France

Received  December 2010 Revised  March 2012 Published  May 2012

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.
Citation: Faker Ben Belgacem. Uniqueness for an ill-posed reaction-dispersion model. Application to organic pollution in stream-waters. Inverse Problems & Imaging, 2012, 6 (2) : 163-181. doi: 10.3934/ipi.2012.6.163
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