|
On a two-point free boundary problem
Author(s):
Jong-Shenq
Guo;
Bei
Hu
Journal:
Quart. Appl. Math.
64
(2006),
413-431.
MSC (2000):
Primary 35K20, 35K55
Posted:
June 12, 2006
Retrieve article in:
PDF DVI PostScript
Abstract |
References |
Similar articles |
Additional information
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.
References:
-
- 1.
- J. D. Buckmaster and G. S. S. Ludford, Theory of Laminar Flames, Cambridge University Press, Cambridge, 1982. MR 0666866 (84f:80011)
- 2.
- 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. MR 1260199 (95e:35097)
- 3.
- 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. MR 1993377 (2005a:35288)
- 4.
- 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. MR 1992860 (2004f:35193)
- 5.
- A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, NJ, 1964. MR 0181836 (31:6062)
- 6.
- 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. MR 1472115 (98h:35238)
- 7.
- Y. Giga and R. V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure and Applied Math. 38 (1985), 297-319. MR 0784476 (86k:35065)
- 8.
- J.-S. Guo, and B. Hu, Quenching profile for a quasilinear parabolic equation, Quarterly of Applied Math. 58 (2000), 613-626. MR 1788421 (2001m:35155)
- 9.
- 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. MR 1927223
- 10.
- D. Hilhorst and J. Hulshof, A free boundary focusing problem, Proc. Amer. Math. Soc. 121 (1994), 1193-1202. MR 1233975 (94j:35200)
- 11.
- Y. Kohsaka, Free boundary problem for quasilinear parabolic equation with fixed angle of contact to a boundary, Nonlinear Analysis 45 (2001), 865-894. MR 1845031 (2002j:35320)
- 12.
- 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. MR 1462990 (98h:35246)
- 13.
- 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. MR 0223758 (36:6806)
Similar Articles:
Retrieve articles in Quarterly of Applied Mathematics
with MSC
(2000):
35K20, 35K55
Retrieve articles in all Journals with MSC
(2000):
35K20, 35K55
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
PII:
S0033-569X-06-01021-1
Received by editor(s):
January 18, 2005
Posted:
June 12, 2006
Copyright of article:
Copyright
2006,
Brown University
The copyright for this article reverts to public domain after 28 years from publication.
|
