Ghanmi, Chifaa; Aouadi, Saloua Mani; Triki, Faouzi Recovering the initial condition in the one-phase Stefan problem. (English) Zbl 1487.35446 Discrete Contin. Dyn. Syst., Ser. S 15, No. 5, 1143-1164 (2022). Summary: We consider the problem of recovering the initial condition in the one-dimensional one-phase Stefan problem for the heat equation from the knowledge of the position of the melting point. We first recall some properties of the free boundary solution. Then we study the uniqueness and stability of the inversion. The principal contribution of the paper is a new logarithmic type stability estimate that shows that the inversion may be severely ill-posed. The proof is based on integral equations representation techniques, and the unique continuation property for parabolic type solutions. We also present few numerical examples operating with noisy synthetic data. Cited in 2 Documents MSC: 35R30 Inverse problems for PDEs 35K20 Initial-boundary value problems for second-order parabolic equations 35R35 Free boundary problems for PDEs 80A22 Stefan problems, phase changes, etc. 45Q05 Inverse problems for integral equations 65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs Keywords:inverse Stefan problem; free boundary problem; heat equation; stability analysis; integral equation; Tikhonov regularization method; one space dimension × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] K. Ammari; A. Bchatnia; K. El Mufti, A remark on observability of the wave equation with moving boundary, J. Appl. Anal., 23, 43-51 (2017) · Zbl 1365.35088 · doi:10.1515/jaa-2017-0007 [2] K. Ammari; F. Triki, On weak observability for evolution systems with skew-adjoint generators, SIAM J. Math. Anal., 52, 1884-1902 (2020) · Zbl 1445.35319 · doi:10.1137/19M1241830 [3] G. Bruckner; J. 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