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I'm looking for some interesting questions and maybe open problems in inverse problems theory, especially in the framework of parabolic PDEs (basically the heat equation). As key words here we can cite for instance:

1) Carleman estimate (to prove uniqueness and stability results).

2) Regularization theory (to regularize ill-posedness).

If someone knows any interesting questions or open problems it would be helpful to list it here. Any recommendation of surveys papers is welcome.

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Inverse Problems for Partial Differential Equations (third edition, 2017) by Victor Isakov concludes each chapter with a collection of open research problems. Chapter 9 is specifically devoted to inverse parabolic problems. I reproduce one of the "open problems" from that chapter:

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