Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

B-method approach to blow-up solutions of fourth-order semilinear parabolic equations

  • 15 Accesses

Abstract

B-method is a novel method developed by Beck et al. (SIAM J. Sci. Comput. 37(5), A2998–A3029, 2015), and has been shown theoretically to be very advantageous in time discretization of the second-order parabolic equations with blow-up solutions. In this paper, we extend the B-method to approximate the blow-up solution of a class of fourth-order parabolic equations, which plays very important role in many engineering applications. First, by following the systematic means of constructing numerical schemes based on the technique of variation of constants proposed by Beck et al., we give some B-method schemes for the fourth-order semilinear parabolic equations. Second, we perform a truncation error analysis to show when and why the B-method scheme is advantageous over its classical counterpart. Third, we take one of the constructed numerical schemes as an example to show the well-posedness using the technique of upper and lower solutions. Last, we carry out numerical experiments to approximate the blow-up solutions and illustrate the efficiency of our numerical schemes.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Notes

  1. 1.

    The maximal and minimal solutions\(\overline {u}\), \(\underline {u}\)are in the sense that if uis a solution of (20) in\(\{\hat {u},\tilde {u}\}\), then\(\underline {u}\leq u\leq \overline {u}\).

  2. 2.

    Unlike the second-order elliptic problem, there is no uniform result on the positivity of solutions for fourth-order problem (20). For one-dimensional problem, some sufficient and necessary conditions can be found in [23]. While for high dimensional problems, we refer the reader to [22] for some elementary results.

  3. 3.

    In fact, when x is tending to the boundary Ω, |Δtng(un(x))| could be infinite. However, in application the spatial domain will be meshed and any interior node can not tend to the boundary Ω in a given mesh. Then for ease of notation, we regarded the term |Δtng(un(x))| as bounded in our analysis.

References

  1. 1.

    Ball, J.M.: Finite time blow-up in nonlinear problems. In: Crandall, M.G. (ed.) Nonlinear Evolution Equations, pp 189–205. Academic Press, New York (1978)

  2. 2.

    Beck, M., Gander, M. J., Kwok, F.: B-methods for the numerical solution of evolution problems with blow-up solutions part I: variation of the constant. SIAM J. Sci. Comput. 37(6), A2998–A3029 (2015)

  3. 3.

    Berger, M. J., Kohn, R. V.: A rescaling algorithm for the numerical calculation of blowing-up solutions. Commun. Pure Appl. Math. 41(6), 841–863 (1988)

  4. 4.

    Bertozzi, A. L., Pugh, M. C.: Finite-time blow-up of solutions of some long-wave unstable thin film equations. Indiana University Mathematics Journal 49 (4), 1323–1366 (2000)

  5. 5.

    Budd, C. J.: Adaptivity with moving grids. Acta Numerica 18(18), 111–241 (2009)

  6. 6.

    Budd, C. J., Huang, W., Russell, R. D.: Moving mesh methods for problems with blow-up. SIAM J. Sci. Comput. 17(2), 305–327 (1996)

  7. 7.

    Budd, C. J., Williams, J. F., Galaktionov, V. A.: Self-similar blow-up in higher-order semilinear parabolic equations. SIAM J. Appl. Math. 64(5), 1775–1809 (2004)

  8. 8.

    Ceniceros, H. D., Hou, T. Y.: An efficient dynamically adaptive mesh for potentially singular solutions. J. Comput. Phys. 172(2), 609–639 (2001)

  9. 9.

    Chaves, M.: Regional blow-up for a higher-order semilinear parabolic equation. Eur. J. Appl. Math. 12(5), 601–623 (2001)

  10. 10.

    Elliott, C. M., Songmu, Z.: On the Cahn-Hilliard equation. Arch. Ration. Mech. Anal. 96(4), 339–357 (1986)

  11. 11.

    Frank-Kamenetskii, D.A.: Towards temperature distributions in a reaction vessel and the stationary theory of thermal explosion. Doklady Acad. Nauk SSSR 18, 411–412 (1938)

  12. 12.

    Friedman, A., Oswald, L.: The blow-up time for higher order semilinear parabolic equations with small leading coefficients. Journal of Differential Equations 75(2), 239–263 (1988)

  13. 13.

    Galaktionov, V. A.: Five types of blow-up in a semilinear fourth-order reaction-diffusion equation: an analytic-numerical approach. Nonlinearity 22(7), 1695–1741 (2009)

  14. 14.

    Galaktionov, V. A., Williams, J. F.: Blow-up in a fourth-order semilinear parabolic equation from explosion-convection theory. Eur. J. Appl. Math. 14(6), 745–764 (2003)

  15. 15.

    Gander, M. J., Liu, Y.: On the definition of Dirichlet and Neumann conditions for the biharmonic equation and its impact on associated Schwarz methods. Domain Decomposition Methods in Science and Engineering XXIII, pp 303–311. Springer, Cham (2017)

  16. 16.

    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, volume 31. Springer Science & Business Media (2006)

  17. 17.

    Huang, W., Ma, J., Russell, R. D.: A study of moving mesh PDE methods for numerical simulation of blowup in reaction diffusion equations. J. Comput. Phys. 227(13), 6532–6552 (2008)

  18. 18.

    Le Roux, M. N.: Semidiscretization in time of nonlinear parabolic equations with blowup of the solution. SIAM J. Numer. Anal. 31(1), 170–195 (1994)

  19. 19.

    Levine, H. A.: Some nonexistence and instability theorems for solutions of formally parabolic equations of the form \({P}u_{t}=-{A}u+\mathcal {F}(u)\). Arch. Ration. Mech. Anal. 51(5), 371–386 (1973)

  20. 20.

    Pao, C. V.: On fourth-order elliptic boundary value problems. Proceedings of the American Mathematical Society 128(4), 1023–1030 (2000)

  21. 21.

    Rakotoson, J. E., Rakotoson, J. M., Verbeke, C.: Generalized lubrification models blow-up and global existence result. RACSAM 99(2), 235–241 (2005)

  22. 22.

    Sato, T., Watanabe, T.: Singular positive solutions for a fourth order elliptic problem in R. Communications on Pure & Applied Analysis 10(1), 245–268 (2011)

  23. 23.

    Shi, G., Chen, S.: Positive solutions of fourth-order superlinear singular boundary value problems. Bulletin of the Australian Mathematical Society 66(1), 95–104 (2002)

  24. 24.

    Van Den Berg, J. B., Vandervorst, R. C.: Stable patterns for fourth-order parabolic equations. Duke Mathematical Journal 115(3), 513–558 (2002)

  25. 25.

    Wang, Y.: On fourth-order elliptic boundary value problems with nonmonotone nonlinear function. J. Math. Anal. Appl. 307(1), 1–11 (2005)

Download references

Acknowledgments

We thank the anonymous referees for their constructive suggestions and useful comments, which have substantially improved our paper.

Funding

Yongkui Zou is supported by NSFC-11771179 and NSFC-91630201. Yingxiang Xu is supported by NSFC-11671074 and the Fundamental Research Funds for the Central Universities (No. 2412018ZD001)

Author information

Correspondence to Yingxiang Xu.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Huo, G., Zou, Y. & Xu, Y. B-method approach to blow-up solutions of fourth-order semilinear parabolic equations. Numer Algor (2020). https://doi.org/10.1007/s11075-019-00868-7

Download citation

Keywords

  • Fourth-order parabolic equation
  • B-method
  • Blow-up solution