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

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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.

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Notes

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

  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. In fact, when x is tending to the boundary ∂Ω, |Δtn − g(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 |Δtn − g(un(x))| as bounded in our analysis.

References

  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. 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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  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)

    Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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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)

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Authors and Affiliations

  1. School of Mathematics, Jilin University, Changchun, 130012, China

    Guanze Huo & Yongkui Zou

  2. School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, China

    Yingxiang Xu

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  1. Guanze Huo
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  2. Yongkui Zou
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Correspondence to Yingxiang Xu.

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Huo, G., Zou, Y. & Xu, Y. B-method approach to blow-up solutions of fourth-order semilinear parabolic equations. Numer Algor 85, 1365–1384 (2020). https://doi.org/10.1007/s11075-019-00868-7

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