On a two-point free boundary problem
Authors:
Jong-Shenq Guo and Bei Hu
Journal:
Quart. Appl. Math. 64 (2006), 413-431
MSC (2000):
Primary 35K20, 35K55
DOI:
https://doi.org/10.1090/S0033-569X-06-01021-1
Published electronically:
June 12, 2006
MathSciNet review:
2259046
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Abstract: We study a two-point free boundary problem for a quasilinear parabolic equation. This problem arises in the model of flame propagation in combustion theory. It also arises in the study of the motion of interface moving with curvature in which the studied problem is confined in the conical region bounded by two straight lines and the interface has prescribed touching angles with these two straight lines. Depending on these two touching angles, there are three different cases, namely, area-expanding, area-preserving, and area-shrinking cases. We first give a proof of the global existence in the expanding and preserving cases. Then the convergence to a line in the preserving case is derived. Finally, in the shrinking case, we show the finite-time vanishing and the convergence of the solution to a self-similar solution.
- J. D. Buckmaster and G. S. S. Ludford, Theory of laminar flames, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, Cambridge-New York, 1982. Electronic & Electrical Engineering Research Studies: Pattern Recognition & Image Processing Series, 2. MR 666866
- Luis A. Caffarelli and Juan L. Vázquez, A free-boundary problem for the heat equation arising in flame propagation, Trans. Amer. Math. Soc. 347 (1995), no. 2, 411–441. MR 1260199, DOI https://doi.org/10.1090/S0002-9947-1995-1260199-7
- Ya-Ling Chang, Jong-Shenq Guo, and Yoshihito Kohsaka, On a two-point free boundary problem for a quasilinear parabolic equation, Asymptot. Anal. 34 (2003), no. 3-4, 333–358. MR 1993377
- Hua-Huai Chern, Jong-Shenq Guo, and Chu-Pin Lo, The self-similar expanding curve for the curvature flow equation, Proc. Amer. Math. Soc. 131 (2003), no. 10, 3191–3201. MR 1992860, DOI https://doi.org/10.1090/S0002-9939-03-07055-2
- Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
- Victor A. Galaktionov, Josephus Hulshof, and Juan L. Vazquez, Extinction and focusing behaviour of spherical and annular flames described by a free boundary problem, J. Math. Pures Appl. (9) 76 (1997), no. 7, 563–608. MR 1472115, DOI https://doi.org/10.1016/S0021-7824%2897%2989963-1
- Yoshikazu Giga and Robert V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math. 38 (1985), no. 3, 297–319. MR 784476, DOI https://doi.org/10.1002/cpa.3160380304
- Jong-Sheng Guo and Bei Hu, Quenching profile for a quasilinear parabolic equation, Quart. Appl. Math. 58 (2000), no. 4, 613–626. MR 1788421, DOI https://doi.org/10.1090/qam/1788421
- Jong-Shenq Guo and Yoshihito Kohsaka, Self-similar solutions of two-point free boundary problem for heat equation, Sūrikaisekikenkyūsho K\B{o}kyūroku 1258 (2002), 94–107. Nonlinear diffusive systems and related topics (Japanese) (Kyoto, 2001). MR 1927223
- Danielle Hilhorst and Josephus Hulshof, A free boundary focusing problem, Proc. Amer. Math. Soc. 121 (1994), no. 4, 1193–1202. MR 1233975, DOI https://doi.org/10.1090/S0002-9939-1994-1233975-9
- Yoshihito Kohsaka, Free boundary problem for quasilinear parabolic equation with fixed angle of contact to a boundary, Nonlinear Anal. 45 (2001), no. 7, Ser. A: Theory Methods, 865–894. MR 1845031, DOI https://doi.org/10.1016/S0362-546X%2899%2900422-8
- J. L. Vazquez, The free boundary problem for the heat equation with fixed gradient condition, Free boundary problems, theory and applications (Zakopane, 1995) Pitman Res. Notes Math. Ser., vol. 363, Longman, Harlow, 1996, pp. 277–302. MR 1462990
- T. I. Zelenjak, Stabilization of solutions of boundary value problems for a second-order parabolic equation with one space variable, Differencial′nye Uravnenija 4 (1968), 34–45 (Russian). MR 0223758
BL J. D. Buckmaster and G. S. S. Ludford, Theory of Laminar Flames, Cambridge University Press, Cambridge, 1982.
CV L. A. Caffarelli and J. L. Vazquez, A free-boundary problem for the heat equation arising in flame propagation, Trans. Amer. Math. Soc. 347 (1995), 411-441.
CGK Y.-L. Chang, J.-S. Guo, and Y. Kohsaka, On a two-point free boundary problem for a quasilinear parabolic equation, Asymptotic Analysis 34 (2003), 333-358.
CGL H.-H. Chern, J.-S. Guo and C.-P. Lo, The self-similar expanding curve for the curvature flow equation, Proc. Amer. Math. Soc. 131 (2003), 3191-3201.
F A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, NJ, 1964.
GHV V. A. Galaktionov, J. Hulshof and J. L. Vazquez, Extinction and focusing behaviour of spherical and annular flames described by a free boundary problem, J. Math. Pures Appl. 76 (1997), 563-608.
GiK Y. Giga and R. V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure and Applied Math. 38 (1985), 297-319.
GH J.-S. Guo, and B. Hu, Quenching profile for a quasilinear parabolic equation, Quarterly of Applied Math. 58 (2000), 613-626.
GK J.-S. Guo and Y. Kohsaka, Self-similar solutions of two-point free boundary problem for heat equation, Nonlinear Diffusive Systems and Related Topics, RIMS Kokyuroku 1258, Research Institute for Mathematical Sciences, Kyoto University, April, 2002, pp. 94-107.
HH2 D. Hilhorst and J. Hulshof, A free boundary focusing problem, Proc. Amer. Math. Soc. 121 (1994), 1193-1202.
K Y. Kohsaka, Free boundary problem for quasilinear parabolic equation with fixed angle of contact to a boundary, Nonlinear Analysis 45 (2001), 865-894.
V J. L. Vazquez, The free boundary problem for the heat equation with fixed gradient condition, Free boundary problems, theory and applications, Zakopane, Poland (1995), Pitman Res. Notes in Math. Ser. 363, 277-302.
Z T. I. Zelenjak, Stabilization of solutions of boundary value problems for a second-order parabolic equation with one space variable, Differential Equations 4 (1968), 17-22.
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Additional Information
Jong-Shenq Guo
Affiliation:
Department of Mathematics, National Taiwan Normal University, S-4 Ting Chou Road, Taipei 117, Taiwan
Email:
jsguo@math.ntnu.edu.tw
Bei Hu
Affiliation:
Department of Mathematics, University of Notre Dame, Room 255, Hurley, Notre Dame, Indiana 46556
Email:
b1hu@nd.edu
Received by editor(s):
January 18, 2005
Published electronically:
June 12, 2006
Article copyright:
© Copyright 2006
Brown University
The copyright for this article reverts to public domain 28 years after publication.
