Chinese Annals of Mathematics, Series B

, Volume 40, Issue 2, pp 309–320 | Cite as

Nonexistence of Type II Blowup for Heat Equation with Exponential Nonlinearity

  • Ruihong Ji
  • Shan LiEmail author
  • Hui Chen


This paper deals with the blowup behavior of the radially symmetric solution of the nonlinear heat equation ut = Δu + eu in ℝN. The authors show the nonexistence of type II blowup under radial symmetric case in the lower supercritical range 3 ≤ N ≤ 9, and give a sufficient condition for the occurrence of type I blowup. The result extends that of Fila and Pulkkinen (2008) in a finite ball to the whole space.


Nonlinear heat equation Type II blowup Exponential nonlinearity 

2000 MR Subject Classification

35K55 35B44 35K05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The authors would like to thank the referees for their helpful comments on the manuscript.


  1. [1]
    Bebernes, J. and Eberly, D., Mathematical Problems from Combustion Theory, Applied Mathematical Sciences, 83, Springer-Verlag, New York, 1989.Google Scholar
  2. [2]
    Chen, X. Y. and Poláçik, P., Asymptotic periodicity of positive solutions of reaction diffusion equations on a ball, J. Reine Angew. Math., 472, 1996, 17–51.MathSciNetzbMATHGoogle Scholar
  3. [3]
    Escauriaza, L., Seregin, G. and Śverák, V., Backward uniqueness for parabolic equations, Arch. Ration. Mech. Anal., 169, 2003, 147–157.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Fila, M., Blowup Solutions of Supercritical Parabolic Equations, Handbook of Differential Equations II, Elsevier, Amsterdam, 2005, 105–158.zbMATHGoogle Scholar
  5. [5]
    Fila, M. and Pulkkinen, A., Nonconstant selfsimilar blowup profile for the exponential reaction-diffusion equation, Tohoku Math. J., 60, 2008, 303–328.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Friedman, A. and McLeod, J. B., Blowup of positive solutions of semilinear heat equations, Indiana Univ. Math. J., 34, 1985, 425–447.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Galaktionov, V. A. and Vázquez, J. L., The problem of blowup in nonlinear parabolic equations, Disc. Contin. Dyn. Systems, 8(2), 2002, 399–433.CrossRefzbMATHGoogle Scholar
  8. [8]
    Gel’fand, I. M., Some problems in the theory of quasilinear equations, Amer. Math. Soc. Transl., 29(2), 1963, 295–381.MathSciNetzbMATHGoogle Scholar
  9. [9]
    Giga, Y., Matsui, S. and Sasayama, S., On blowup rate for sign-changing solutions in a convex domain, Math. Meth. Appl. Sci., 27, 2004, 1771–1782.CrossRefzbMATHGoogle Scholar
  10. [10]
    Giga, Y., Matsui, S. and Sasayama, S., Blowup rate for semilinear heat equation with subcritical nonlinearity, Indiana Univ. Math. J., 53, 2004, 483–514.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Herrero, M. A. and Vázquez, J. L., Explosion de solutions des équations paraboliques semilinéaires supercritiques, C. R. Acad. Sci. Paris, Sér. I Math., 319, 1994, 141–145.Google Scholar
  12. [12]
    Herrero, M. A. and V´azquez, J. L., A blowup result for semilinear heat equations in the supercritical case, preprint.Google Scholar
  13. [13]
    Joseph, D. D. and Lundgren, T. S., Quasilinear Dirichlet problems driven by positive sources, Arch. Ration. Mech. Anal., 49, 1973, 241–269.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Kazdan, J. L. and Warner, F. W., Curvature functions for compact 2-manifolds, Ann. Math., 99, 1974, 14–47.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Lacey, A. A., Global, unbouded solutions to a parabolic equation, J. Differential Equations, 101, 1993, 80–102.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Matano, H., Blowup in nonlinear heat equations with supercritical power nonlinearity, Contem. Math., 446, 2007, 385–412.CrossRefzbMATHGoogle Scholar
  17. [17]
    Matano, H. and Merle, F., On non-existence of type II blow up for a supercritical nonlinear heat equation, Comm. Pure Appl. Math., 57, 2004, 1494–1541.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Matano, H. and Merle, F., Classification of type I and type II behaviors for a supercritical nonlinear heat equation, J. Func. Anal., 256, 2009, 992–1064.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Mizoguchi, N., Type II blowup for a semilinear heat equation, Adv. Diff. Equ., 9, 2004, 1279–1316.MathSciNetzbMATHGoogle Scholar
  20. [20]
    Mizoguchi, N., Blowup behavior of solutions for a semilinear heat equation with supercritical nonlinearity, J. Differential Equations, 205, 2004, 298–328.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    Mizoguchi, N., Boundedness of global solutions for a semilinear heat equation with supercritical nonlinearity, Indiana Univ. Math. J., 54(4), 2005, 1047–1059.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Mizoguchi, N., Nonexistence of type II blowup solution for a semilinear heat equation, J. Differential Equations, 250, 2011, 26–32.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    Peral, I. and Vázquez, J. L., On the stability or instability of the singular solution of the semilinear heat equation with exponential reaction term, Arch. Rat. Mech. Anal., 129, 1995, 201–224.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Samarskii, A. A., Galaktionov, V. A., Kurdyumov, S. P. and Mikhailov, A. P., Blowup in Quasilinear Parabolic Equations, Walter de Gruyter, Berlin, New York, 1995.CrossRefzbMATHGoogle Scholar

Copyright information

© Fudan University and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Geomathematics Key Laboratory of Sichuan ProvinceChengdu University of TechnologyChengduChina
  2. 2.Business SchoolSichuan UniversityChengduChina

Personalised recommendations