The quenching behavior of a semilinear heat equation with a singular boundary outflux

Selcuk, Burhan; Ozalp, Nuri

doi:10.1090/S0033-569X-2014-01367-9

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
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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

C. Y. Chan, Recent advances in quenching phenomena, Proceedings of Dynamic Systems and Applications, Vol. 2 (Atlanta, GA, 1995) Dynamic, Atlanta, GA, 1996, pp. 107–113. MR 1419518
C. Y. Chan, New results in quenching, World Congress of Nonlinear Analysts ’92, Vol. I–IV (Tampa, FL, 1992) de Gruyter, Berlin, 1996, pp. 427–434. 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, 553–568. MR 2086046, DOI https://doi.org/10.1090/qam/2086046
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. 127–137. MR 1372958, 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, 203–209. MR 1830870, 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, 574–584. MR 1709705, DOI https://doi.org/10.1007/s000330050167
Keng Deng and Cheng-Lin Zhao, Blow-up versus quenching, Commun. Appl. Anal. 7 (2003), no. 1, 87–100. MR 1954906
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, 892–901. MR 2386469, 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, 795–802. MR 1246508, 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, 565–581. MR 2017226
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, 729–736. 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, 1123–1135. 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, 53–72. 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, 343–356. 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, 77–88. Advances in quenching. MR 1820667
Timo Salin, On quenching with logarithmic singularity, Nonlinear Anal. 52 (2003), no. 1, 261–289. 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, 333–339. 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, 955–959. 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, 624–630. MR 2294876, DOI https://doi.org/10.1016/j.amc.2006.06.061