On a twopoint free boundary problem
Authors:
JongShenq Guo and Bei Hu
Journal:
Quart. Appl. Math. 64 (2006), 413431
MSC (2000):
Primary 35K20, 35K55
Published electronically:
June 12, 2006
MathSciNet review:
2259046
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Abstract: We study a twopoint 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, areaexpanding, areapreserving, and areashrinking 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 finitetime vanishing and the convergence of the solution to a selfsimilar solution.
 1.
J.
D. Buckmaster and G.
S. S. Ludford, Theory of laminar flames, Cambridge Monographs
on Mechanics and Applied Mathematics, Cambridge University Press,
Cambridge, 1982. Electronic & Electrical Engineering Research Studies:
Pattern Recognition & Image Processing Series, 2. MR 666866
(84f:80011)
 2.
Luis
A. Caffarelli and Juan
L. Vázquez, A freeboundary problem for the heat
equation arising in flame propagation, Trans.
Amer. Math. Soc. 347 (1995), no. 2, 411–441. MR 1260199
(95e:35097), http://dx.doi.org/10.1090/S00029947199512601997
 3.
YaLing
Chang, JongShenq
Guo, and Yoshihito
Kohsaka, On a twopoint free boundary problem for a quasilinear
parabolic equation, Asymptot. Anal. 34 (2003),
no. 34, 333–358. MR 1993377
(2005a:35288)
 4.
HuaHuai
Chern, JongShenq
Guo, and ChuPin
Lo, The selfsimilar expanding curve for
the curvature flow equation, Proc. Amer. Math.
Soc. 131 (2003), no. 10, 3191–3201 (electronic). MR 1992860
(2004f:35193), http://dx.doi.org/10.1090/S0002993903070552
 5.
Avner
Friedman, Partial differential equations of parabolic type,
PrenticeHall Inc., Englewood Cliffs, N.J., 1964. MR 0181836
(31 #6062)
 6.
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
(98h:35238), http://dx.doi.org/10.1016/S00217824(97)899631
 7.
Yoshikazu
Giga and Robert
V. Kohn, Asymptotically selfsimilar blowup of semilinear heat
equations, Comm. Pure Appl. Math. 38 (1985),
no. 3, 297–319. MR 784476
(86k:35065), http://dx.doi.org/10.1002/cpa.3160380304
 8.
JongSheng
Guo and Bei
Hu, Quenching profile for a quasilinear parabolic equation,
Quart. Appl. Math. 58 (2000), no. 4, 613–626.
MR
1788421 (2001m:35155)
 9.
JongShenq
Guo and Yoshihito
Kohsaka, Selfsimilar solutions of twopoint free boundary problem
for heat equation, Sūrikaisekikenkyūsho
Kōkyūroku 1258 (2002), 94–107.
Nonlinear diffusive systems and related topics (Japanese) (Kyoto, 2001). MR
1927223
 10.
Danielle
Hilhorst and Josephus
Hulshof, A free boundary focusing
problem, Proc. Amer. Math. Soc.
121 (1994), no. 4,
1193–1202. MR 1233975
(94j:35200), http://dx.doi.org/10.1090/S00029939199412339759
 11.
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
(2002j:35320), http://dx.doi.org/10.1016/S0362546X(99)004228
 12.
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
(98h:35246)
 13.
T.
I. Zelenjak, Stabilization of solutions of boundary value problems
for a secondorder parabolic equation with one space variable,
Differencial′nye Uravnenija 4 (1968), 34–45
(Russian). MR
0223758 (36 #6806)
 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 freeboundary problem for the heat equation arising in flame propagation, Trans. Amer. Math. Soc. 347 (1995), 411441. MR 1260199 (95e:35097)
 3.
 Y.L. Chang, J.S. Guo, and Y. Kohsaka, On a twopoint free boundary problem for a quasilinear parabolic equation, Asymptotic Analysis 34 (2003), 333358. MR 1993377 (2005a:35288)
 4.
 H.H. Chern, J.S. Guo and C.P. Lo, The selfsimilar expanding curve for the curvature flow equation, Proc. Amer. Math. Soc. 131 (2003), 31913201. MR 1992860 (2004f:35193)
 5.
 A. Friedman, Partial Differential Equations of Parabolic Type, PrenticeHall, 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), 563608. MR 1472115 (98h:35238)
 7.
 Y. Giga and R. V. Kohn, Asymptotically selfsimilar blowup of semilinear heat equations, Comm. Pure and Applied Math. 38 (1985), 297319. MR 0784476 (86k:35065)
 8.
 J.S. Guo, and B. Hu, Quenching profile for a quasilinear parabolic equation, Quarterly of Applied Math. 58 (2000), 613626. MR 1788421 (2001m:35155)
 9.
 J.S. Guo and Y. Kohsaka, Selfsimilar solutions of twopoint 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. 94107. MR 1927223
 10.
 D. Hilhorst and J. Hulshof, A free boundary focusing problem, Proc. Amer. Math. Soc. 121 (1994), 11931202. 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), 865894. 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, 277302. MR 1462990 (98h:35246)
 13.
 T. I. Zelenjak, Stabilization of solutions of boundary value problems for a secondorder parabolic equation with one space variable, Differential Equations 4 (1968), 1722. MR 0223758 (36:6806)
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Additional Information
JongShenq Guo
Affiliation:
Department of Mathematics, National Taiwan Normal University, S4 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
DOI:
http://dx.doi.org/10.1090/S0033569X06010211
PII:
S 0033569X(06)010211
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.
