A numerical method of estimating blow-up rates for nonlinear evolution equations by using rescaling algorithm

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Abstract

In this paper, we proposed a numerical method for estimating blow-up rate of blow-up solutions for a class of nonlinear evolution equations which have a scaling invariance. To use this scaling invariance we adopt the rescaling algorithm to the problems and numerically estimate the blow-up rates. Applying the method to several examples, we examine the effectiveness of the method.

Keywords

Blow-up rate Rescaling algorithm Numerical method 

Mathematics Subject Classification

65L99 65M99 

Notes

Acknowledgements

The second author is partially supported by JSPS KAKENHI Grant Numbers 15K13461 and 15H03632. The authors thank the referees for careful reading and invaluable comments on our manuscript.

References

  1. 1.
    Anada, K., Ishiwata, T.: Blow-up rates of solutions of initial-boundary value problems for a quasi-linear parabolic equation. J. Differ. Equ. 262, 181–271 (2017)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Andrews, B.: Singularities in crystalline curvature flows. Asian J. Math. 6(1), 101–122 (2002)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Berger, M., Kohn, R.V.: A rescaling algorithm for the numerical calculation of blowing-up solutions. Commun. Pure Appl. Math. 41, 841–863 (1988)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Budd, C.J., Huang, W.-Z., Russell, R.D.: Moving mesh methods for problems with blow-up. SIAM J. Sci. Comput. 17, 305–327 (1996)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Chen, Y.G.: Asymptotic behaviors of blowing-up solutions for finite difference analogue of \(u_t = u_{xx} + u^{1+\alpha }\). J. Fac. Sci. Univ. Tokyo Sect. IA Math. 33, 541–574 (1986)MathSciNetGoogle Scholar
  6. 6.
    Cho, C.-H.: On the computation of the numerical blow-up time. Jpn. J. Ind. Appl. Math. 30, 331–349 (2013)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cho, C.-H.: Numerical detection of blow-up: a new sufficient condition for blow-up. Jpn. J. Ind. Appl. Math. 33(1), 81–98 (2016)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chou, K.-S., Zhu, X.-P.: The Curve Shortening Problem. Chapman and Hall/CRC, Boca Raton (2001)CrossRefMATHGoogle Scholar
  9. 9.
    Hirota, C., Ozawa, K.: Numerical method of estimating the blow-up time and rate of the solution of ordinary differential equations an application to the blow-up problems of partial differential equations. J. Comput. Appl. Math. 193(2), 614–637 (2006)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Horstmann, D.: From 1970 until present: the Keller–Segel model in chemotaxis and its consequences. I. Jahresber. Deutsch. Math.-Verein. 105(3), 103–165 (2003)MathSciNetMATHGoogle Scholar
  11. 11.
    Horstmann, D.: From 1970 until present: the Keller–Segel model in chemotaxis and its consequences. II. Jahresber. Deutsch. Math.-Verein. 106(2), 51–69 (2004)MathSciNetMATHGoogle Scholar
  12. 12.
    Ishiwata, T., Yazaki, S.: On the blow-up rate for fast blow-up solutions arising in an anisotropic crystalline motion. J. Comput. Appl. Math. 159, 55–64 (2003)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Nakagawa, T.: Blowing up of a finite difference solution to \(u_t = u_{xx} + u^2\). Appl. Math. Optim. 2, 337–350 (1976)CrossRefMATHGoogle Scholar
  14. 14.
    Quittner, P., Souplet, P.: Superlinear Parabolic Problems, Blow-Up, Global Existence and Steady States. Birkhäuser Verlag AG, Basel (2007)MATHGoogle Scholar
  15. 15.
    Saito, N., Sasaki, T.: Finite difference approximation for nonlinear Schrödinger equations with application to blow-up computation. Jpn. J. Ind. Appl. Math. 33(2), 427–470 (2016)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Samarskii, A.A., Galaktionov, V.A., Kurdyumov, S.P., Mikhailov, A.P.: Blow-Up in Quasilinear Parabolic Equations. Walter de Gruyter, Berlin (1995)Google Scholar
  17. 17.
    Takayasu, A., Matsue, K., Sasaki, T., Tanaka, K., Mizuguchi, M., Oishi, S.: Numerical validation of blow-up solutions of ordinary differential equations. J. Comput. Appl. Math. 314, 10–29 (2017)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Ushijima, T.: On the approximation of blow-up time for solutions of nonlinear parabolic equations. Publ. RIMS Kyoto Univ. 36, 613–640 (2000)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Zhou, G., Saito, N.: Finite volume methods for a Keller–Segel system: discrete energy, error estimates and numerical blow-up analysis. Numer. Math. 135, 265–311 (2017)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© The JJIAM Publishing Committee and Springer Japan KK 2017

Authors and Affiliations

  1. 1.Waseda University Senior High SchoolNerima-kuJapan
  2. 2.Department of Mathematical SciencesShibaura Institute of TechnologyMinuma-kuJapan
  3. 3.Deparment of Mathematics, Faculty of Science and TechnologyTokyo University of ScienceNodaJapan

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