Open AccessRecovering the initial condition in the one-phase Stefan problem
Author(s) -
Chifaa Ghanmi,
Saloua Mani Aouadi,
Faouzi Triki
Publication year - 2022
Publication title -
discrete and continuous dynamical systems. series s
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.481
H-Index - 34
eISSN - 1937-1632
pISSN - 1937-1179
DOI - 10.3934/dcdss.2021087
Subject(s) - uniqueness , stefan problem , mathematics , continuation , heat equation , boundary value problem , mathematical analysis , inversion (geology) , free boundary problem , well posed problem , logarithm , boundary (topology) , computer science , paleontology , structural basin , biology , programming language
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.