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Non-simultaneous blow up for coupled nonlinear diffusion equations with absorptions

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Abstract

This paper deals with coupled nonlinear diffusion equations with absorptions. We characterize the range of parameters for which non-simultaneous blow up occurs. We establish the necessary and sufficient conditions for the occurrence of non-simultaneous blow up with proper initial data. Moreover, we obtain the optimal condition under which any blow up is non-simultaneous.

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References

  1. Bräandle C., Quirós F., Rossi J.D.: Non-simultaneous blow-up for a quasilinear parabolic system with reaction at boundary. Commun. Pure Appl. Anal. 4, 523–536 (2005)

    Article  MathSciNet  Google Scholar 

  2. Deng K., Xu M.: Remarks on blow-up behavior for a nonlinear diffusion equation with Neumann boundary conditions. Proc. Am. Math. Soc. 127, 167–172 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Du L.L., Yao Z.A.: Note on non-simultaneous blow-up for a reaction-diffusion system. Appl. Math. Lett. 21, 81–85 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Filo J.: Diffusivity versus absorption through the boundary. J. Differ. Equ. 99, 281–305 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. Friedman A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, NJ (1964)

    MATH  Google Scholar 

  6. Ladyzhenskaya, O.A., Solonnikov, V.A., Uralceva, N.N.: Linear and quasilinear equations of parabolic type. In: Transl. Math. Mono., vol. 23, AMS, Providence RI (1968)

  7. Lieberman G.M.: Second Order Parabolic Differential Equations. World Scientific, River Edge (1996)

    Book  MATH  Google Scholar 

  8. Liu B.C., Liu F.J.: Non-simultaneous blow-up of n components for nonlinear parabolic systems. J. Math. Anal. Appl. 356, 215–231 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Pao C.V.: Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York, London (1992)

    MATH  Google Scholar 

  10. Quirós F., Rossi J.D.: Non-simultaneous blow-up in a nonlinear parabolic system. Adv. Nonlinear Stud. 3, 397–418 (2003)

    MathSciNet  MATH  Google Scholar 

  11. Smoller J.: Shock Waves and Reaction-Diffusion Equations. Springer, New York (1983)

    Book  MATH  Google Scholar 

  12. Song X.F., Zheng S.N.: Multi-nonlinear interactions in quasilinear reaction-diffusion equations with nonlinear boundary flux. Math. Comput. Model. 39, 133–144 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Souplet P., Tayachi S.: Optimal condition for non-simultaneous blow-up in a reaction-diusion system. J. Math. Soc. Japan 56, 571–584 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Zheng S.N., Liu B.C.: A nonlinear diffusion system with convection. Nonlinear Anal. 63, 123–135 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Zheng S.N., Qiao L.: Non-simultaneous blow-up in a reaction-diffusion system. Appl. Math. Comput. 180, 309–317 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Zheng S.N., Song X.F.: Interactions among multi-nonlinearities in a nonlinear diffusion system with absorptions and nonlinear boundary flux. Nonlinear Anal. 57, 519–530 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zheng S.N., Wang W.: Blow-up rate for a nonlinear diffusion equation. Appl. Math. Lett. 19, 1385–1389 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ziemer W.P.: Interior and boundary continuity of weak solutions of degenerate parabolic equations. Trans. Am. Math. Soc. 271, 733–748 (1982)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. College of mathematic and information, China West Normal University, Nanchong, 637009, People’s Republic of China

    Zhongping Li & Wanjuan Du

  2. College of mathematics and Statistics, Chongqing University, Chongqing, 400044, People’s Republic of China

    Chunlai Mu

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  1. Zhongping Li
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  2. Chunlai Mu
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  3. Wanjuan Du
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Correspondence to Zhongping Li.

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Li, Z., Mu, C. & Du, W. Non-simultaneous blow up for coupled nonlinear diffusion equations with absorptions. Z. Angew. Math. Phys. 64, 253–263 (2013). https://doi.org/10.1007/s00033-012-0234-8

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  • DOI: https://doi.org/10.1007/s00033-012-0234-8

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