Identification of a time-dependent point source in a linear transport equation with spatially varying coefficients: detection of pollution source

One motivation for our study concerns an environmental application that regards the identification of pollution sources in surface water: in a river, for example, the introduction of organic matter which could have as origin city sewages, industrial wastes,... usually drops to too low the level of the dissolved oxygen in the water. Since the lack of dissolved oxygen represents a serious threat to the diversity of the acquatic life, then localizing pollution sources and recovering the history of the loaded organic matter could play a crucial role in preventing worse consequences regarding the perish of many acquatic species as well as in alerting downstream drinking water stations about the presence of an accidental pollution. This can be done by recording the BOD (Biological Oxygen Demand) concentration which represents the amount of dissolved oxygen required by the micro-organism living in the river to decompose the introduced organic substances [1]. Therefore, the more organic material there is, the higher the BOD concentration. Besides, in a portion of a river represented by a segment (0, `) which we suppose controlling during a time T , the BOD concentration denoted here by u is governed in Q = (0, `)× (0, T ) by the following one-dimensional parabolic partial differential equation appended to initial and boundary conditions, for more details see [3, 5]: