The quenching behavior of a semilinear heat equation with a singular boundary outflux. (English) Zbl 1327.35207
Summary: 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.
MSC:
| 35K61 | Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations |
| 35B35 | Stability in context of PDEs |
| 35K58 | Semilinear parabolic equations |
| 35K67 | Singular parabolic equations |
Keywords:
semilinear heat equation; singular boundary outflux; quenching; quenching point; quenching time; maximum principlesReferences:
| [1] | 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. · Zbl 0908.35056 |
| [2] | 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. · Zbl 0847.35063 |
| [3] | 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. · Zbl 1072.35100 · doi:10.1090/qam/2086046 |
| [4] | 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. · Zbl 0845.35047 · doi:10.1142/9789812796417_0010 |
| [5] | 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. · Zbl 1027.35053 · doi:10.1016/S0096-3003(99)00278-7 |
| [6] | 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. · Zbl 0929.35012 · doi:10.1007/s000330050167 |
| [7] | Keng Deng and Cheng-Lin Zhao, Blow-up versus quenching, Commun. Appl. Anal. 7 (2003), no. 1, 87 – 100. · Zbl 1085.35069 |
| [8] | 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. · Zbl 1142.35047 · doi:10.1016/j.jmaa.2007.05.077 |
| [9] | Marek Fila and Howard A. Levine, Quenching on the boundary, Nonlinear Anal. 21 (1993), no. 10, 795 – 802. · Zbl 0809.35043 · doi:10.1016/0362-546X(93)90124-B |
| [10] | 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. · Zbl 1052.35101 |
| [11] | Hideo Kawarada, On solutions of initial-boundary problem for \?_{\?}=\?\?\?+1/(1-\?), Publ. Res. Inst. Math. Sci. 10 (1974/75), no. 3, 729 – 736. · Zbl 0306.35059 · doi:10.2977/prims/1195191889 |
| [12] | L. Ke and S. Ning, Quenching for degenerate parabolic equations, Nonlinear Anal. 34 (1998), no. 7, 1123 – 1135. · Zbl 0941.35006 · doi:10.1016/S0362-546X(98)00039-X |
| [13] | C. M. Kirk and Catherine A. Roberts, A quenching problem for the heat equation, J. Integral Equations Appl. 14 (2002), no. 1, 53 – 72. · Zbl 1050.35039 · doi:10.1216/jiea/1031315434 |
| [14] | 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). · Zbl 1030.35099 |
| [15] | 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. · Zbl 0985.45004 |
| [16] | Timo Salin, On quenching with logarithmic singularity, Nonlinear Anal. 52 (2003), no. 1, 261 – 289. · Zbl 1020.35004 · doi:10.1016/S0362-546X(02)00110-4 |
| [17] | 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. · Zbl 1238.35042 · doi:10.1016/j.nonrwa.2011.07.040 |
| [18] | Ying Yang, Jingxue Yin, and Chunhua Jin, A quenching phenomenon for one-dimensional \?-Laplacian with singular boundary flux, Appl. Math. Lett. 23 (2010), no. 9, 955 – 959. · Zbl 1195.35193 · doi:10.1016/j.aml.2010.04.001 |
| [19] | 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. · Zbl 1113.35104 · doi:10.1016/j.amc.2006.06.061 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
