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


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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  1. 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. 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.


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


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