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On the role of conservation laws and input data in the generation of peaking modes in quasilinear multidimensional parabolic equations with nonlinear source and in their approximations

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Abstract

We study unbounded solutions of a broad class of initial–boundary value problems for multidimensional quasilinear parabolic equations with a nonlinear source. By using a conservation law, we obtain conditions imposed solely on the input data and ensuring that a solution of the problem blows up in finite time. The blow-up time of the solution is estimated from above. By approximating the source function with the use of Steklov averaging with weight function coordinated with the nonlinear coefficients of the elliptic operator, we construct finite-difference schemes satisfying a grid counterpart of the integral conservation law.

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Authors and Affiliations

  1. Institute of Mathematics, National Academy of Sciences, Minsk, Belarus

    P. P. Matus, N. G. Churbanova & D. A. Shchadinskii

  2. The John Paul II Catholic University of Lublin, Lublin, Poland

    P. P. Matus, N. G. Churbanova & D. A. Shchadinskii

  3. Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, Russia

    P. P. Matus, N. G. Churbanova & D. A. Shchadinskii

Authors
  1. P. P. Matus
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  2. N. G. Churbanova
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  3. D. A. Shchadinskii
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Correspondence to P. P. Matus.

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Original Russian Text © P.P. Matus, N.G. Churbanova, D.A. Shchadinskii, 2016, published in Differentsial’nye Uravneniya, 2016, Vol. 52, No. 7, pp. 981–989.

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Matus, P.P., Churbanova, N.G. & Shchadinskii, D.A. On the role of conservation laws and input data in the generation of peaking modes in quasilinear multidimensional parabolic equations with nonlinear source and in their approximations. Diff Equat 52, 942–950 (2016). https://doi.org/10.1134/S0012266116070120

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  • DOI: https://doi.org/10.1134/S0012266116070120

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