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The main purpose of this work is to study an inverse source problem for degenerate/singular parabolic equations with degeneracy and singularity occurring in the interior of the spatial domain. Using Carleman estimates, we prove a Lipschitz stability estimate for the source term provided that additional measurement data are given on a suitable interior subdomain. For the numerical solution, the reconstruction is formulated as a minimization problem using the output least squares approach with the Tikhonov regularization. The Fréchet differentiability of the Tikhonov functional and the Lipschitz continuity of the Fréchet gradient are proved. These properties allow us to apply gradient methods for numerical solution of the considered inverse source problem.

In this article, we study the null controllability of population model with an interior degenerate diffusion. To this end, we proved first a new Carleman estimate for the full adjoint system and then we deduce a suitable observability inequality which will be needed to establish the existence of a control acting on a subset of the space which lead the population to extinction in a finite time.

In this article, we study an inverse problem for linear degenerate parabolic systems with one force. We establish Lipschitz stability for the source term from measurements of one component of the solution at a positive time and on a subset of the space domain, which contains degeneracy points. The key ingredient is the derivation of a Carleman-type estimate.

In this article, we study an inverse problem for weakly degenerate coupled parabolic systems with one force. We establish Lipschitz stability for the source term from measurements of one component of the solution at a positive time and on a subset of the space domain. The key ingredient is the derivation of a Carleman-type estimate.