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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access April 1, 2015

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

  • Rossitza Semerdjieva
From the journal Open Mathematics

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.

References

[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. Search in Google Scholar

[2] J. R. Cannon, The one-dimensional heat equation, Encyclopedia of Mathematics and its Applications, vol. 23, Addison-Wesley, Menlo Park, 1984. Search in Google Scholar

[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. 10.1016/0022-247X(68)90180-7Search in Google Scholar

[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. 10.1007/BF02415069Search in Google Scholar

[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. Search in Google Scholar

[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. Search in Google Scholar

[7] S. Cui, A. Friedman, Analysis of a mathematical model of the growth of necrotic tumors, J. Math. Anal. Appl. 255 (2001), 636– 677. 10.1006/jmaa.2000.7306Search in Google Scholar

[8] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice–Hall, Englewood Cliffs, N.J., 1964. Search in Google Scholar

[9] A. Friedman, F. Reitich, Analysis of a mathematical model for the growth of tumors, J. Math. Biol. 38 (1999), 262–284. 10.1007/s002850050149Search in Google Scholar

[10] A. Friedman, Mathematical analysis and challenges arising from models of tumor growth. Math. Models Methods Appl. Sci. 17 (2007), suppl., 1751–1772. Search in Google Scholar

[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. Search in Google Scholar

[12] L. I. Rubinstein, The Stefan problem. Translations of Mathematical Monographs, Vol. 27. AMS, Providence, R.I., 1971. Search in Google Scholar

[13] D. Schaeffer, A new proof of infinite differentiability of the free boundary in the Stefan problem, J. Diff. Equat. 20 (1976), 266–269. 10.1016/0022-0396(76)90106-6Search in Google Scholar

[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. Search in Google Scholar

Received: 2014-1-22
Accepted: 2015-3-18
Published Online: 2015-4-1

©2015 Rossitza Semerdjieva

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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