On the smoothness of the free boundary in a nonlocal one-dimensional parabolic free boundary value problem

• Autorzy: Rossitza Semerdjieva
• Języki publikacji: EN

Abstrakty

EN

We consider one-dimensional parabolic free boundary value problem with a nonlocal (integro-differential) condition on the free boundary. Results on Cm-smoothness of the free boundary are obtained. In particular, a necessary and sufficient condition for infinite differentiability of the free boundary is given.

Słowa kluczowe

EN

Free boundary problem; Parabolic equation; Mixed type boundary conditions; Nonlocal condition; Smoothness of the free boundary

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2015
• Tom: 13
• Numer: 1

Daty

otrzymano

2014-01-22

zaakceptowano

2015-03-18

online

2015-04-01

Twórcy

autor

Rossitza Semerdjieva

• Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G.
  Bonchev Str., Bl. 8, 1113 Sofia, Bulgaria

Bibliografia

• [1] N. Bellomo, N. K. Li, P. K. Maini, On the foundations of cancer modelling: selected topics, speculations, and perspectives. Math. Models Methods Appl. Sci. 18 (2008), no. 4, 593–646. [WoS][Crossref]
• [2] J. R. Cannon, The one-dimensional heat equation, Encyclopedia of Mathematics and its Applications, vol. 23, Addison-Wesley, Menlo Park, 1984.
• [3] J. R. Cannon and C. D. Hill, On the infinite differentiability of the free boundary in a Stefan problem, J. Math. Anal. Appl. 22 (1968), 385–397. [Crossref]
• [4] J. R. Cannon and M. Primicerio, A two phase Stefan problem: regularity of the free boundary, Ann. Mat. Pure. Appl. 88 (1971), 217–228.
• [5] Chiang Li-Shang, Existence and differentiability of the solution of the two-phase Stefan problem for quasilinear parabolic equations, Chinese Math.–Acta. 7 (1965), 481–496.
• [6] A. Corli, V. Guidi and M. Primicerio, On a diffusion problem arising in nanophased thin films, Adv. Math. Sci. Appl. 18 (2008), 517– 533.
• [7] S. Cui, A. Friedman, Analysis of a mathematical model of the growth of necrotic tumors, J. Math. Anal. Appl. 255 (2001), 636– 677. [Crossref]
• [8] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice–Hall, Englewood Cliffs, N.J., 1964.
• [9] A. Friedman, F. Reitich, Analysis of a mathematical model for the growth of tumors, J. Math. Biol. 38 (1999), 262–284. [Crossref]
• [10] A. Friedman, Mathematical analysis and challenges arising from models of tumor growth. Math. Models Methods Appl. Sci. 17 (2007), suppl., 1751–1772. [Crossref][WoS]
• [11] O. A. Ladyzhenskaja, V. A. Solonnikov and N. N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, Vol. 23, Amer. Math. Soc., Providence, R.I., 1968.
• [12] L. I. Rubinstein, The Stefan problem. Translations of Mathematical Monographs, Vol. 27. AMS, Providence, R.I., 1971.
• [13] D. Schaeffer, A new proof of infinite differentiability of the free boundary in the Stefan problem, J. Diff. Equat. 20 (1976), 266–269. [Crossref]
• [14] R. Semerdjieva, Global existence of classical solutions for a nonlocal one dimensional parabolic free boundary problem, Houston J. Math. 40 (2014), no. 1, 229–253.

Identyfikatory

DOI

10.1515/math-2015-0026