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We consider the transport equation in where is a bounded domain, and discuss two inverse problems which consist of determining a vector-valued function or a real-valued function by initial values and data on a subboundary of . Our results are conditional stability of Hölder type in a subdomain D provided that the outward normal component of is positive on . The proofs are based on a Carleman estimate where the weight function depends on H. [ABSTRACT FROM AUTHOR]

In this paper, we prove Lipschitz stability results for inverse source problems relative to parabolic equations. We use the method introduced by Imanuvilov and Yamamoto in 1998 based on Carleman estimates. What is new here is that we study a class of one-dimensional degenerate parabolic equations. In our model, the diffusion coefficient vanishes at one extreme point of the domain. Instead of the classical Carleman estimates obtained by Fursikov and Imanuvilov for non degenerate equations, we use and extend some recent Carleman estimates for degenerate equations obtained by Cannarsa, Martinez and Vancostenoble. Finally, we obtain Lipschitz stability results in inverse source problems for our class of degenerate parabolic equations both in the case of a boundary observation and in the case of a locally distributed observation. [ABSTRACT FROM AUTHOR]