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

Rossitza Semerdjieva

Open Mathematics (2015)

  • Volume: 13, Issue: 1, page 229-253
  • ISSN: 2391-5455

Abstract

top
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.

How to cite

top

Rossitza Semerdjieva. "On the smoothness of the free boundary in a nonlocal one-dimensional parabolic free boundary value problem." Open Mathematics 13.1 (2015): 229-253. <http://eudml.org/doc/269966>.

@article{RossitzaSemerdjieva2015,
abstract = {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.},
author = {Rossitza Semerdjieva},
journal = {Open Mathematics},
keywords = {Free boundary problem; Parabolic equation; Mixed type boundary conditions; Nonlocal condition; Smoothness of the free boundary; parabolic equation; nonlocal free boundary problem; global classical solution},
language = {eng},
number = {1},
pages = {229-253},
title = {On the smoothness of the free boundary in a nonlocal one-dimensional parabolic free boundary value problem},
url = {http://eudml.org/doc/269966},
volume = {13},
year = {2015},
}

TY - JOUR
AU - Rossitza Semerdjieva
TI - On the smoothness of the free boundary in a nonlocal one-dimensional parabolic free boundary value problem
JO - Open Mathematics
PY - 2015
VL - 13
IS - 1
SP - 229
EP - 253
AB - 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.
LA - eng
KW - Free boundary problem; Parabolic equation; Mixed type boundary conditions; Nonlocal condition; Smoothness of the free boundary; parabolic equation; nonlocal free boundary problem; global classical solution
UR - http://eudml.org/doc/269966
ER -

References

top
  1. [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] Zbl1151.92014
  2. [2] J. R. Cannon, The one-dimensional heat equation, Encyclopedia of Mathematics and its Applications, vol. 23, Addison-Wesley, Menlo Park, 1984. Zbl0567.35001
  3. [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] Zbl0167.10504
  4. [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. Zbl0219.35048
  5. [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. [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. Zbl1183.35157
  7. [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] Zbl0984.35169
  8. [8] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice–Hall, Englewood Cliffs, N.J., 1964. Zbl0144.34903
  9. [9] A. Friedman, F. Reitich, Analysis of a mathematical model for the growth of tumors, J. Math. Biol. 38 (1999), 262–284. [Crossref] Zbl0944.92018
  10. [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] Zbl1135.92013
  11. [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. [12] L. I. Rubinstein, The Stefan problem. Translations of Mathematical Monographs, Vol. 27. AMS, Providence, R.I., 1971. 
  13. [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] Zbl0314.35044
  14. [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. Zbl1301.35221

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.