Identification of a boundary influx condition in a one-phase Stefan problem

References

• Goldman NL. Inverse Stefan problem. Dordrecht: Kluwer Academic Publishers; 1997.
   Google Scholar
• Rubinstein LI. The Stefan problem. Providence (RI): American Mathematical Society; 1971.
   Google Scholar
• Meirmanov AM. The Stefan problem. Berlin: De Gruyter; 1992.
   Google Scholar
• Visintin A. Models of phase transitions. Boston (MA): Birkhäuser; 1996. (Progress in Nonlinear Differential Equations and Their Applications; 28).
   Google Scholar
• Friedman A. Variational principles and free-Boundary problems. New York (NY): Wiley-Interscience; 1982.
   Google Scholar
• Ladyzenskaya OA, Solonnikov VA, Ural'ceva NN. Linear and quasilinear equations of parabolic type. Vol. 23, Transl. [Math. Monographs. Amer. Math. Soc.]. 1968.
   Google Scholar
• DiBenedetto E, Friedman A. The ill-posed Hele-Shaw model and the Stefan problem for supercooled water. Trans Am Math Soc. 1984;282:183–204.
   Web of Science ®Google Scholar
• Pruss J, Saal J, Simonett G. Existence of analytic solutions for the classical Stefan problem. Math Ann. 2007;338(3):703–755.
   Web of Science ®Google Scholar
• Colton D. The inverse Stefan problem for the heat equation in two space variables. Mathematika. 1974;21:282–286.
   Web of Science ®Google Scholar
• Johansson BT, Lesnic D, Reeve T. A method of fundamental solutions for the one-dimensional inverse Stefan problem. Appl Math Model. 2011;35(9):4367–4378.
   Web of Science ®Google Scholar
• Jochum P. The numerical solution of the inverse Stefan problem. Numer Math. 1980;34:411–429.
   Web of Science ®Google Scholar
• Jochum P. To the solution of an inverse Stefan problem in two space variables, In: Albrecht J, editor. Numerical treatment of free boundary value problems. Basel: Birkhäuser-Verlag; 1982. p. 127–136. ISNM 58.
   Google Scholar
• Reemtsen R, Kirsch A. A method for the numerical solution of the one dimensional inverse Stefan problem. Numerische Mathematik. 1984;45(2):253–273.
   Web of Science ®Google Scholar
• Wrobel LC. A boundary element solution to Stefan's problem. In: Brebbia CA, Futagami T, Tanaka M, editors. Boundary elements V. Berlin: Computational Mechanics Publications, Springer-Verlag; 1983. p. 173–182.
   Google Scholar
• Cannon JR, Douglas Jr J. The stability of the boundary in a stefan problem. Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze. 1967;21 6 (1):83–91.
   Google Scholar
• Wei T, Yamamoto M. Reconstruction of a moving boundary from Cauchy data in one-dimensional heat equation. Inverse Probl Sci Eng. 2009;17:551–567.
   Web of Science ®Google Scholar
• Cannon JR. The one-dimensional heat equation. Vol. 23, Cambridge University Press; 1984.
   Google Scholar
• Friedman A. Partial differential equations of parabolic type. Englewood Cliffs (NJ): Prentice-Hall Inc.; 1964.
   Google Scholar
• Sacks PE. Continuity of solutions of a singular parabolic equation. Nonlinear Anal. 1983;7:387–409.
   Web of Science ®Google Scholar
• Cannon JR, Hill CD. Existence, uniqueness, stability, and monotone dependence in a Stefan problem for the heat equation. J Math Mech. 1967;17:1–20.
   Web of Science ®Google Scholar
• Cannon JR, Douglas J. The Cauchy problem for the heat equation. SIAM J Numer Anal. 1967;4(3):317–336.
   Google Scholar
• Kyner WT. An existence and uniqueness theorem for non-linear Stefan problem. Math Mech. 1959;8:483–498.
   Google Scholar
• Cannon JR, Mario P. Remarks on the one-phase Stefan problem for the heat equation with the flux prescribed on the fixed boundary. J Math Anal Appl. 1971;35(2):361–373.
   Web of Science ®Google Scholar
• Evans LC. Partial differential equations. Vol. 19, American Mathematical Society; 2010.
   Google Scholar
• Ammari K, Triki F. On weak Observability for evolution systems with Skew-Adjoint generators. SIAM J Math Anal. 2020;52(2):1884–1902.
   Web of Science ®Google Scholar
• Rudolf G, Yamamoto M. Operator theoretic treatment of linear Abel integral equations of first kind. Jpn J Ind Appl Math. 1999;16(1):137–161.
   Web of Science ®Google Scholar
• Brunner H. Volterra integral equations: an introduction to theory and applications. Vol. 30, Cambridge University Press; 2017.
   Google Scholar
• Choulli M, Yamamoto M. Logarithmic stability of parabolic Cauchy problems. preprint (2017) arXiv:1702.06299.
   Google Scholar
• Bao G, Triki F. Stability for the multifrequency inverse medium problem. J Differ Equ. 2020;269(9):7106–7128.
   Web of Science ®Google Scholar
• Nevanlinna R. Analytic Functions. Berlin Heidelberg: Springer; [translated from german by B. Eckmann]. 1970.
   Google Scholar
• Alessandrini G, Alberto F. Sharp stability estimates of harmonic continuation along lines. Math Methods Appl Sci. 2000;23:1037–1056.
   Web of Science ®Google Scholar
• Landau HG. Heat conduction in a melting solid. Quart Appl Math. 1950;8(1):81–94.
   Web of Science ®Google Scholar
• Smith GD. Numerical solution of partial differential equations: finite difference methods. Oxford university press; 1985.
   Google Scholar
• Wang YF, Zhang Y, Lukyanenko DV, et al. Recovering aerosol particle size distribution function on the set of bounded piecewise-convex functions. Inverse Probl Sci Eng. 2013;21(2):339–354.
   Web of Science ®Google Scholar
• Andreucci D. Lecture notes on the Stefan problem. Università da Roma La Sapienza; 2004. (Lecture notes).
   Google Scholar
• Knabner P. Control of Stefan problems by means of linear-quadratic defect minimization. Numer Math. 1985;46(3):429–442.
   Web of Science ®Google Scholar