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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.

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A degenerate parabolic identification problem: the Hilbertian case

A. Lorenzi1

1Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, Via Saldini 50, 20133 Milano, Italy. Email:

Citation Information: Journal of Inverse and Ill-posed Problems. Volume 16, Issue 9, Pages 849–872, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: 10.1515/JIIP.2008.053, December 2008

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We recover a non-negative time-dependent function, vanishing only at t = 0, in a linear multi-dimensional parabolic equation with a non-integrable degeneration concentrated at t = 0. First we prove a global in time existence and uniqueness result for the direct problem in a general Hilbert space, via Fourier representation of the solution to the direct problem. Then we show a similar result, but local in time only, for the identification problem. Finally, we apply such a result to our specific linear parabolic equation related to a smooth bounded domain in ℝd, d = 1, 2, 3.

Key words.: First-order singular in time differential equations in Hilbert space; global well-posedness of the direct problem; identifying a scalar time dependent coefficient in front of the space operator; a local in time well-posedness result for the identification problem; applications to degenerate in time differential parabolic equations

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