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Asymptotic properties of blow-up solutions in reaction–diffusion equations with nonlocal boundary flux

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Abstract

This paper deals with a reaction–diffusion problem with coupled nonlinear inner sources and nonlocal boundary flux. Firstly, we propose the critical exponents on nonsimultaneous blow-up under some conditions on the initial data. Secondly, we combine the scaling technique and the Green’s identity method to determine four kinds of simultaneous blow-up rates. Thirdly, the lower and the upper bounds of blow-up time are derived by using Sobolev-type differential inequalities.

Keywords

Reaction–diffusion system Nonsimultaneous blow-up Blow-up rate Blow-up time 

Mathematics Subject Classification

35K55 35B40 35K15 35B33 

Notes

Acknowledgements

The authors would like to thank the anonymous Referees and the Editor for valuable suggestions improving the first version of this paper. This paper is supported by NNSF of China; Shandong Provincial Natural Science Foundation, China (ZR2016AM12, ZR2017LA003); the Fundamental Research Funds for the Central Universities.

References

  1. 1.
    An, X.W., Song, X.F.: The lower bound for the blowup time of the solution to a quasi-linear parabolic problem. Appl. Math. Lett. 69, 82–86 (2017)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bebernes, J., Eberly, D.: Mathematical Problems from Combustion Theory. Springer, New York (1989)CrossRefMATHGoogle Scholar
  3. 3.
    Brändle, C., Quirós, F., Rossi, J.D.: Non-simultaneous blow-up for a quasilinear parabolic system with reaction at the boundary. Commun. Pure Appl. Math. 4, 523–536 (2004)MathSciNetMATHGoogle Scholar
  4. 4.
    Brändle, C., Quirós, F., Rossi, J.D.: The role of nonlinear diffusion in non-simultaneous blow-up. J. Math. Anal. Appl. 308, 92–104 (2005)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Fu, S.C., Guo, J.S., Tsai, J.C.: Blow-up behavior for a semilinear heat equation with a nonlinear boundary condition. Tohoku Math. J. 55, 565–581 (2003)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Gladkov, A.L., Kavitova, T.V.: Initial-boundary-value problem for a semilinear parabolic equation with nonlinear nonlocal boundary conditions. Ukrain. Math. J. 68, 1–14 (2016)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Gladkov, A.L., Kavitova, T.V.: Blow-up problem for semilinear heat equation with nonlinear nonlocal boundary condition. Appl. Anal. 95, 1974–1988 (2016)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Gómez, J.L., Márquez, C., Wolanski, N.: Blow up results and localization of blow up points for the heat equation with a nonlinear boundary condition. J. Differ. Equ. 92, 384–401 (1991)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Hu, B., Yin, H.M.: The profile near blow up time for solution of the heat equation with a nonlinear boundary condition. Trans. Am. Math. Soc. 346, 117–135 (1994)CrossRefMATHGoogle Scholar
  10. 10.
    Ladyženskaja, O.A., Sol’onnikov, V.A., Uralceva, N.N.: Linear and Quasi-linear Equations of Parabolic Type, vol. 23. Amer. Math. Soc. Transl. (2), Providence (1968)CrossRefGoogle Scholar
  11. 11.
    Li, F.J., Liu, B.C., Zheng, S.N.: Simultaneous and non-simultaneous blow-up for heat equations with coupled nonlinear boundary fluxes. Z. Angew. Math. Phys. 58, 717–735 (2007)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Li, H., Wang, M.X.: Uniform blow-up profiles and boundary layer for a parabolic system with localized nonlinear reaction terms. Sci. China A 48, 185–197 (2005)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Lieberman, G.M.: Second Order Parabolic Differential Equations. World Scientific, River Edge (1996)CrossRefMATHGoogle Scholar
  14. 14.
    Lin, Z.G., Wang, M.X.: The blow-up properties of solutions to semilinear heat equations with nonlinear boundary conditons. Z. Angew. Math. Phys. 50, 361–374 (1999)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Liu, B.C., Li, F.J., Zheng, S.N.: Critical non-simultaneous blow-up exponents for a reaction–diffusion system. Adv. Math. (China) 4, 531–536 (2011)MathSciNetGoogle Scholar
  16. 16.
    Marras, M., Piro, S.V.: Blow-up time estimates in nonlocal reaction–diffusion systems under various boundary conditions. J. Inequal. Appl. 2014, 167 (2014)CrossRefMATHGoogle Scholar
  17. 17.
    Marras, M., Vernier-piro, S.: Reaction–diffusion problems under non-local boundary conditions with blow-up solutions. Bound. Value Probl. 2017, 2 (2017)CrossRefMATHGoogle Scholar
  18. 18.
    Ortoleva, P., Ross, J.: Local structures in chemical reactions with hetergeneous catalysis. J. Chem. Phys. 56, 4397–4400 (1972)CrossRefGoogle Scholar
  19. 19.
    Pao, C.V.: Nonlinear Parabolic and Elliptic Equations. Plenum, New York (1992)MATHGoogle Scholar
  20. 20.
    Payne, L.E., Philippin, G.A., Vernier-Piro, S.: Blow up phenomena for a semilinear heat equation with nonlinear boundary condiion. Nonlinear Anal. 73, 971–978 (2010)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Pinasco, J.P., Rossi, J.D.: Simultaneous versus non-simultaneous blow-up. N. Z. J. Math. 29, 55–59 (2000)MathSciNetMATHGoogle Scholar
  22. 22.
    Quirós, F., Rossi, J.D.: Non-simultaneous blow-up in a semilinear parabolic system. Z. Angew. Math. Phys. 52, 342–346 (2001)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Rossi, J.D., Souplet, P.: Coexistence of simultaneous and non-simultaneous blow-up in a semilinear parabolic system. Differ. Integral Equ. 18, 405–418 (2005)MATHGoogle Scholar
  24. 24.
    Souplet, Ph, Tayachi, S.: Blow up rates for nonlinear heat equations with gradient terms and for parabolic inequalities. Colloq. Math. 88, 135–154 (2001)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Souplet, Ph, Tayachi, S.: Optimal condition for non-simultaneous blow-up in a reaction–diffusion system. J. Math. Soc. Jpn. 56, 571–584 (2004)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Wang, M.X.: Blow-up estimates for a semilinear reaction diffusion system. J. Math. Anal. Appl. 257, 46–51 (2001)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Wang, M.X.: Blow-up rate estimates for semilinear parabolic systems. J. Differ. Equ. 170, 317–324 (2001)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Wang, M.X.: Blow-up properties of solutions to parabolic systems coupled in equations and boundary conditions. J. Lond. Math. Soc. 67, 180–194 (2003)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Zheng, S.N., Liu, B.C., Li, F.J.: Blow-up rate estimates for a doubly coupled reaction–diffusion system. J. Math. Anal. Appl. 312, 576–595 (2005)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Zheng, S.N., Liu, B.C., Li, F.J.: Non-simultaneous blow-up for a multi-coupled reaction–diffusion system. Nonlinear Anal. 64, 1189–1202 (2006)MathSciNetCrossRefMATHGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of ScienceChina University of PetroleumQingdaoPeople’s Republic of China

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