The quenching behavior of a semilinear heat equation with a singular boundary outflux
Authors:
Burhan Selcuk and Nuri Ozalp
Journal:
Quart. Appl. Math. 72 (2014), 747-752
MSC (2000):
Primary 35K55, 35K60, 35B35, 35Q60
DOI:
https://doi.org/10.1090/S0033-569X-2014-01367-9
Published electronically:
September 26, 2014
MathSciNet review:
3291826
Full-text PDF Free Access
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Additional Information
Abstract: In this paper, we study the quenching behavior of the solution of a semilinear heat equation with a singular boundary outflux. We prove a finite-time quenching for the solution. Further, we show that quenching occurs on the boundary under certain conditions and we show that the time derivative blows up at a quenching point. Finally, we get a quenching rate and a lower bound for the quenching time.
References
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- Hideo Kawarada, On solutions of initial-boundary problem for $u_{t}=u_{xx}+1/(1-u)$, Publ. Res. Inst. Math. Sci. 10 (1974/75), no. 3, 729β736. MR 0385328, DOI https://doi.org/10.2977/prims/1195191889
- L. Ke and S. Ning, Quenching for degenerate parabolic equations, Nonlinear Anal. 34 (1998), no. 7, 1123β1135. MR 1637229, DOI https://doi.org/10.1016/S0362-546X%2898%2900039-X
- C. M. Kirk and Catherine A. Roberts, A quenching problem for the heat equation, J. Integral Equations Appl. 14 (2002), no. 1, 53β72. MR 1932536, DOI https://doi.org/10.1216/jiea/1031315434
- C. M. Kirk and Catherine A. Roberts, A review of quenching results in the context of nonlinear Volterra equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 10 (2003), no. 1-3, 343β356. Second International Conference on Dynamics of Continuous, Discrete and Impulsive Systems (London, ON, 2001). MR 1974255
- W. E. Olmstead and Catherine A. Roberts, Critical speed for quenching, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 8 (2001), no. 1, 77β88. Advances in quenching. MR 1820667
- Timo Salin, On quenching with logarithmic singularity, Nonlinear Anal. 52 (2003), no. 1, 261β289. MR 1938660, DOI https://doi.org/10.1016/S0362-546X%2802%2900110-4
- Runzhang Xu, Chunyan Jin, Tao Yu, and Yacheng Liu, On quenching for some parabolic problems with combined power-type nonlinearities, Nonlinear Anal. Real World Appl. 13 (2012), no. 1, 333β339. MR 2846843, DOI https://doi.org/10.1016/j.nonrwa.2011.07.040
- Ying Yang, Jingxue Yin, and Chunhua Jin, A quenching phenomenon for one-dimensional $p$-Laplacian with singular boundary flux, Appl. Math. Lett. 23 (2010), no. 9, 955β959. MR 2659118, DOI https://doi.org/10.1016/j.aml.2010.04.001
- Yuanhong Zhi and Chunlai Mu, The quenching behavior of a nonlinear parabolic equation with nonlinear boundary outflux, Appl. Math. Comput. 184 (2007), no. 2, 624β630. MR 2294876, DOI https://doi.org/10.1016/j.amc.2006.06.061
References
- C. Y. Chan, Recent advances in quenching phenomena, Proceedings of Dynamic Systems and Applications, Vol. 2 (Atlanta, GA, 1995), Dynamic, Atlanta, GA, 1996, pp. 107β113. MR 1419518 (98a:35065)
- C. Y. Chan, New results in quenching, World Congress of Nonlinear Analysts β92, Vol. IβIV (Tampa, FL, 1992), de Gruyter, Berlin, 1996, pp. 427β434. MR 1389093
- C. Y. Chan and X. O. Jiang, Quenching for a degenerate parabolic problem due to a concentrated nonlinear source, Quart. Appl. Math. 62 (2004), no. 3, 553β568. MR 2086046 (2005e:35139)
- C. Y. Chan and N. Ozalp, Singular reaction-diffusion mixed boundary-value quenching problems, Dynamical systems and applications, World Sci. Ser. Appl. Anal., vol. 4, World Sci. Publ., River Edge, NJ, 1995, pp. 127β137. MR 1372958 (97a:35109), DOI https://doi.org/10.1142/9789812796417_0010
- C. Y. Chan and S. I. Yuen, Parabolic problems with nonlinear absorptions and releases at the boundaries, Appl. Math. Comput. 121 (2001), no. 2-3, 203β209. MR 1830870 (2002a:35121), DOI https://doi.org/10.1016/S0096-3003%2899%2900278-7
- Keng Deng and Mingxi Xu, Quenching for a nonlinear diffusion equation with a singular boundary condition, Z. Angew. Math. Phys. 50 (1999), no. 4, 574β584. MR 1709705 (2000e:35110), DOI https://doi.org/10.1007/s000330050167
- Keng Deng and Cheng-Lin Zhao, Blow-up versus quenching, Commun. Appl. Anal. 7 (2003), no. 1, 87β100. MR 1954906 (2003j:35170)
- Nadejda E. Dyakevich, Existence, uniqueness, and quenching properties of solutions for degenerate semilinear parabolic problems with second boundary conditions, J. Math. Anal. Appl. 338 (2008), no. 2, 892β901. MR 2386469 (2009c:35223), DOI https://doi.org/10.1016/j.jmaa.2007.05.077
- Marek Fila and Howard A. Levine, Quenching on the boundary, Nonlinear Anal. 21 (1993), no. 10, 795β802. MR 1246508 (95b:35028), DOI https://doi.org/10.1016/0362-546X%2893%2990124-B
- Sheng-Chen Fu, Jong-Shenq Guo, and Je-Chiang Tsai, Blow-up behavior for a semilinear heat equation with a nonlinear boundary condition, Tohoku Math. J. (2) 55 (2003), no. 4, 565β581. MR 2017226 (2004h:35112)
- Hideo Kawarada, On solutions of initial-boundary problem for $u_{t}=u_{xx}+1/(1-u)$, Publ. Res. Inst. Math. Sci. 10 (1974/75), no. 3, 729β736. MR 0385328 (52 \#6192)
- L. Ke and S. Ning, Quenching for degenerate parabolic equations, Nonlinear Anal. 34 (1998), no. 7, 1123β1135. MR 1637229 (2000b:35138), DOI https://doi.org/10.1016/S0362-546X%2898%2900039-X
- C. M. Kirk and Catherine A. Roberts, A quenching problem for the heat equation, J. Integral Equations Appl. 14 (2002), no. 1, 53β72. MR 1932536 (2003g:35129), DOI https://doi.org/10.1216/jiea/1031315434
- C. M. Kirk and Catherine A. Roberts, A review of quenching results in the context of nonlinear Volterra equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 10 (2003), no. 1-3, 343β356. Second International Conference on Dynamics of Continuous, Discrete and Impulsive Systems (London, ON, 2001). MR 1974255 (2004c:35216)
- W. E. Olmstead and Catherine A. Roberts, Critical speed for quenching, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 8 (2001), no. 1, 77β88. Advances in quenching. MR 1820667 (2002c:35159)
- Timo Salin, On quenching with logarithmic singularity, Nonlinear Anal. 52 (2003), no. 1, 261β289. MR 1938660 (2003j:35182), DOI https://doi.org/10.1016/S0362-546X%2802%2900110-4
- Runzhang Xu, Chunyan Jin, Tao Yu, and Yacheng Liu, On quenching for some parabolic problems with combined power-type nonlinearities, Nonlinear Anal. Real World Appl. 13 (2012), no. 1, 333β339. MR 2846843 (2012i:35205), DOI https://doi.org/10.1016/j.nonrwa.2011.07.040
- Ying Yang, Jingxue Yin, and Chunhua Jin, A quenching phenomenon for one-dimensional $p$-Laplacian with singular boundary flux, Appl. Math. Lett. 23 (2010), no. 9, 955β959. MR 2659118 (2011f:35175), DOI https://doi.org/10.1016/j.aml.2010.04.001
- Yuanhong Zhi and Chunlai Mu, The quenching behavior of a nonlinear parabolic equation with nonlinear boundary outflux, Appl. Math. Comput. 184 (2007), no. 2, 624β630. MR 2294876 (2007k:35249), DOI https://doi.org/10.1016/j.amc.2006.06.061
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Additional Information
Burhan Selcuk
Affiliation:
Department of Computer Engineering, Karabuk University, BalΔ±klarkayasΔ± Mevkii, 78050, Turkey
Email:
bselcuk@karabuk.edu.tr
Nuri Ozalp
Affiliation:
Department of Mathematics, Ankara University, Besevler, 06100, Turkey
Email:
nozalp@science.ankara.edu.tr
Keywords:
Semilinear heat equation,
singular boundary outflux,
quenching,
quenching point,
quenching time,
maximum principles.
Received by editor(s):
November 28, 2012
Received by editor(s) in revised form:
April 23, 2013
Published electronically:
September 26, 2014
Article copyright:
© Copyright 2014
Brown University
