Asymptotic profile of quenching in a system of heat equations coupled at the boundary. (English) Zbl 1273.35055
Authors’ abstract: We study finite time quenching for the radial solutions of a system of heat equations coupled at the boundary conditions. This system exhibits simultaneous and non-simultaneous quenching. In particular, three kinds of simultaneous quenching profiles are obtained for different nonlinear exponent regions and appropriate initial data.
Reviewer: Chiu Yeung Chan (Lafayette)
MSC:
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35K51 | Initial-boundary value problems for second-order parabolic systems |
Keywords:
radial solutions; finite time quenching; quenching set; simultaneous and non-simultaneous quenching; simultaneous quenching profilesReferences:
| [1] | C.Y. Chan and M.K. Kwong: Quenching phenomena for singular nonlinear parabolic equations , Nonlinear Anal. 12 (1988), 1377-1383. · Zbl 0704.35012 · doi:10.1016/0362-546X(88)90085-5 |
| [2] | C.Y. Chan: New results in quenching ; in World Congress of Nonlinear Analysts ’92, 1 - 4 (Tampa, FL, 1992), de Gruyter, Berlin, 1996, 427-434. · Zbl 0847.35063 |
| [3] | J.M. Chadam, A. Peirce and H.-M. Yin: The blowup property of solutions to some diffusion equations with localized nonlinear reactions , J. Math. Anal. Appl. 169 (1992), 313-328. · Zbl 0792.35095 · doi:10.1016/0022-247X(92)90081-N |
| [4] | K. Deng and M. Xu: Quenching for a nonlinear diffusion equation with a singular boundary condition , Z. Angew. Math. Phys. 50 (1999), 574-584. · Zbl 0929.35012 · doi:10.1007/s000330050167 |
| [5] | K. Deng and M. Xu: On solutions of a singular diffusion equation , Nonlinear Anal. 41 (2000), Ser. A: Theory Methods, 489-500. · Zbl 0949.35075 · doi:10.1016/S0362-546X(98)00292-2 |
| [6] | M. Fila and H.A. Levine: Quenching on the boundary , Nonlinear Anal. 21 (1993), 795-802. · Zbl 0809.35043 · doi:10.1016/0362-546X(93)90124-B |
| [7] | R. Ferreira, A. de Pablo, F. Quiros and J.D. Rossi: Non-simultaneous quenching in a system of heat equations coupled at the boundary , Z. Angew. Math. Phys. 57 (2006), 586-594. · Zbl 1101.35008 · doi:10.1007/s00033-005-0003-z |
| [8] | B. Hu and H.-M. Yin: The profile near blowup time for solution of the heat equation with a nonlinear boundary condition , Trans. Amer. Math. Soc. 346 (1994), 117-135. · Zbl 0823.35020 · doi:10.2307/2154944 |
| [9] | H.A. Levine: Advances in quenching ; in Nonlinear Diffusion Equations and Their Equilibrium States, 3 (Gregynog, 1989), Progr. Nonlinear Differential Equations Appl. 7 , Birkhäuser, Boston, Boston, MA, 1992, 319-346. · Zbl 0792.35017 · doi:10.1007/978-1-4612-0393-3_23 |
| [10] | H.A. Levine and G.M. Lieberman: Quenching of solutions of parabolic equations with nonlinear boundary conditions in several dimensions , J. Reine Angew. Math. 345 (1983), 23-38. · Zbl 0519.35041 |
| [11] | H. Kawarada: On solutions of initial-boundary problem for \(u_{t}=u_{xx}+1/(1-u)\) , Publ. Res. Inst. Math. Sci. 10 (1975), 729-736. · Zbl 0306.35059 · doi:10.2977/prims/1195191889 |
| [12] | C.V. Pao: Nonlinear Parabolic and Elliptic Equations, Plenum, New York, 1992. · Zbl 0777.35001 |
| [13] | A. de Pablo, F. Quirós and J.D. Rossi: Nonsimultaneous quenching , Appl. Math. Lett. 15 (2002), 265-269. · Zbl 1019.35036 · doi:10.1016/S0893-9659(01)00128-8 |
| [14] | S. Zheng and X. Song: Quenching rates for heat equations with coupled singular nonlinear boundary flux , Sci. China Ser. A 51 (2008), 1631-1643. · Zbl 1172.35433 · doi:10.1007/s11425-007-0178-1 |
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