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Journal of Evolution Equations

, Volume 17, Issue 2, pp 849–867 | Cite as

Blow-up sets for a complex-valued semilinear heat equation

  • Junichi Harada
Article
  • 96 Downloads

Abstract

This paper is concerned with finite-time blow-up solutions of a one-dimensional complex-valued semilinear heat equation. We characterize the location and the number of blow-up points from the viewpoint of zeros of the solution.

Keywords

System of semilinear parabolic equation Blow-up point 

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References

  1. 1.
    Ackermann N., Bartsch T.: Superstable manifolds of semilinear parabolic problem. J. Dynam. Differential Equations 17(no. 1), 115–173 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cohen P. J., Lees M.: Asymptotic decay of solutions of differential inequalities. Pacific J. Math. 11, 1235–1249 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    P. Constantin, P. D. Lax, and A. Majda A simple one-dimensional model for the three-dimensional vorticity equation, Comm. Pure Appl. Math. 38 no. 6 (1985) 715–724.Google Scholar
  4. 4.
    Filippas S., Cohn R. V.: Refined asymptotics for the blowup of \({u_{t}-\Delta{u}=u^{p}}\). Comm. Pure Appl. Math. 45(no. 7), 821–869 (1992)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Fujita H.: On the blowing up of solutions of the Cauchy problem for \({u_{t}=\Delta{u}+u^{1+\alpha}}\). J. Fac. Sci. Univ. Tokyo Sect. I 13, 109–124 (1966)MathSciNetGoogle Scholar
  6. 6.
    Guo J. S., Ninomiya H., Shimojo M., Yanagida E.: Convergence and blow-up of solutions for a complex-valued heat equation with a quadratic nonlinearity. Trans. Amer. Math. Soc. 365(no. 5), 2447–2467 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Harada J.: Blowup profile for a complex valued semilinear heat equation. J. Funct. Anal. 270(no. 11), 4213–4255 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Herrero M. A., Veláuez J. J. L.: Blow-up profiles in one-dimensional, semilinear parabolic problems. Comm. Partial Differential Equations 17(no. 1–2), 205–219 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Matano H.: Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29(no. 2), 401–441 (1982)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Naito Y., Suzuki T.: Existence of type II blowup solutions for a semilinear heat equation with critical nonlinearity. J. Differential Equations 232((no. 1), 176–211 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ogawa H.: Lower Bounds for Solutions of Differential Inequalities in Hilbert Space. Proc. Amer. Math. Soc. 16, 1241–1243 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    R. Palais Blowup for nonlinear equations using a comparison principle in Fourier space, Comm. Pure Appl. Math. 41 no. 2 (1988) 165–196.Google Scholar
  13. 13.
    Sakajo T.: Blow-up solutions of the Constantin-Lax-Majda equation with a generalized viscosity term. J. Math. Sci. Univ. Tokyo 10(no. 1), 187–207 (2003)MathSciNetzbMATHGoogle Scholar
  14. 14.
    T. Sakajo, On global solutions for the Constantin-Lax-Majda equation with a generalized viscosityterm, Nonlinearity 16 no. 4 (2003) 1319–228.Google Scholar
  15. 15.
    Schochet S.: Explicit solutions of the viscous model vorticity equation. Comm. Pure Appl. Math. 39(no. 4), 531–537 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Yang Y.: Behavior of solutions of model equations for incompressible fluid flow. J. Differential Equations 125(no. 11), 33–153 (1996)MathSciNetGoogle Scholar
  17. 17.
    Nouaili N., Zaag H.: Profile for a simultaneously blowing up solution for a complex valued semilinear heat equation. Comm. Partial Differential Equations 40(no. 7), 1197–1217 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Faculty of Education and Human StudiesAkita UniversityAkitaJapan

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