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Quenching for a Diffusion System with Coupled Boundary Fluxes

Received: 6 January 2016     Accepted: 20 January 2016     Published: 18 February 2016
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Abstract

In this paper, we investigate a diffusion system of two parabolic equations with more general singular coupled boundary fluxes. Within proper conditions, we prove that the finite quenching phenomenon happens to the system. And we also obtain that the quenching is non-simultaneous and the corresponding quenching rate of solutions. This extends the original work by previous authors for a heat system with coupled boundary fluxes subject to non-homogeneous Neumann boundary conditions.

Published in Applied and Computational Mathematics (Volume 5, Issue 1)
DOI 10.11648/j.acm.20160501.13
Page(s) 18-22
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2016. Published by Science Publishing Group

Keywords

Quenching, Quenching Rate, Quenching Point, Singular Term, Parabolic System

References
[1] H. Kawarada. On solutions of initial boundary value problem for , Publ. Res. Inst. Math. Sci. 10(1975), 729-736.
[2] R. Ferreira, A de Pablo, F. Quirs, J. D. Rossi. Non-simultaneous quenching in a system of heat equations coupled at the boundary. Z. Angew. Math. Phys., 57(2006), 586-594.
[3] S. N. Zheng, X. F. Song. Quenching rates for heat equations with coupled nonlinear boundary flux. Sci. China Ser. A. 2008; 51: 1631-1643.
[4] M. Fila, H. A. Levine. Quenching on the boundary. Nonlinear Anal., 21(1993), 795-802.
[5] R. H. Ji, C. Y. Qu, L. D. Wang. Simultaneous and non-simultaneous quenching for coupled parabolic system, Appl. Anal., 94(2), 2015, 233-250.
[6] A. de Pablo, F. Quirós, J. D. Rossi. Non-simultaneous quenching, Appl. Math. Lett. 15 (2002), 265–269.
[7] Y. H. Zhi, C. L. Mu, Non-simultaneous quenching in a semilinear parabolic system with weak singularities of logarithmic type. Applied Mathematics and Computation, 196(2008), 17-23.
[8] R. H. Ji, S. S. Zhou, S. N. Zheng. Quenching behavior of solutions in coupled heat equations with singular multi- nonlinearities, Applied Mathematics and Computation, 223 (2013), 401–410.
[9] C. Y. Chan. Recent advances in quenching phenomena, Proc. Dynam. Systems. Appl. 2(1996), 107-113.
[10] H. A. Levine. The phenomenon of quenching: a survey, in: Trends in the Theory and Practice of NonLinear Analysis, North Holland, New York, 1985, pp. 275-286.
[11] H. A. Levine, J. T. Montgomery. The quenching of solutions of some nonlinear parabolic equations, SIAM J. Math. Anal. 11 (1980), 842-847.
[12] T. Salin. On quenching with logarithmic singularity, Nonlinear Anal. TMA. 52 (2003), 261-289.
[13] C. L. Mu, S. M. Zhou, D. M. Liu. Quenching for a reaction-diffusion system with logarithmic singularity. Nonlinear Anal., 71(2009), 5599-5605.
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  • APA Style

    Haijie Pei, Wenbo Zhao. (2016). Quenching for a Diffusion System with Coupled Boundary Fluxes. Applied and Computational Mathematics, 5(1), 18-22. https://doi.org/10.11648/j.acm.20160501.13

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    ACS Style

    Haijie Pei; Wenbo Zhao. Quenching for a Diffusion System with Coupled Boundary Fluxes. Appl. Comput. Math. 2016, 5(1), 18-22. doi: 10.11648/j.acm.20160501.13

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    AMA Style

    Haijie Pei, Wenbo Zhao. Quenching for a Diffusion System with Coupled Boundary Fluxes. Appl Comput Math. 2016;5(1):18-22. doi: 10.11648/j.acm.20160501.13

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  • @article{10.11648/j.acm.20160501.13,
      author = {Haijie Pei and Wenbo Zhao},
      title = {Quenching for a Diffusion System with Coupled Boundary Fluxes},
      journal = {Applied and Computational Mathematics},
      volume = {5},
      number = {1},
      pages = {18-22},
      doi = {10.11648/j.acm.20160501.13},
      url = {https://doi.org/10.11648/j.acm.20160501.13},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20160501.13},
      abstract = {In this paper, we investigate a diffusion system of two parabolic equations with more general singular coupled boundary fluxes. Within proper conditions, we prove that the finite quenching phenomenon happens to the system. And we also obtain that the quenching is non-simultaneous and the corresponding quenching rate of solutions. This extends the original work by previous authors for a heat system with coupled boundary fluxes subject to non-homogeneous Neumann boundary conditions.},
     year = {2016}
    }
    

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    T1  - Quenching for a Diffusion System with Coupled Boundary Fluxes
    AU  - Haijie Pei
    AU  - Wenbo Zhao
    Y1  - 2016/02/18
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    N1  - https://doi.org/10.11648/j.acm.20160501.13
    DO  - 10.11648/j.acm.20160501.13
    T2  - Applied and Computational Mathematics
    JF  - Applied and Computational Mathematics
    JO  - Applied and Computational Mathematics
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    UR  - https://doi.org/10.11648/j.acm.20160501.13
    AB  - In this paper, we investigate a diffusion system of two parabolic equations with more general singular coupled boundary fluxes. Within proper conditions, we prove that the finite quenching phenomenon happens to the system. And we also obtain that the quenching is non-simultaneous and the corresponding quenching rate of solutions. This extends the original work by previous authors for a heat system with coupled boundary fluxes subject to non-homogeneous Neumann boundary conditions.
    VL  - 5
    IS  - 1
    ER  - 

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Author Information
  • College of Mathematic and Information, China West Norm University, Nanchong, P. R. China

  • College of Mathematic and Information, China West Norm University, Nanchong, P. R. China

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