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Well-posedness of difference schemes for semilinear parabolic equations with weak solutions

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Abstract

The well-posedness of difference schemes approximating initial-boundary value problem for parabolic equations with a nonlinear power-type source is studied. Simple sufficient conditions on the input data are obtained under which the weak solutions of the differential and difference problems are globally stable for all 0 ⩽ t ⩽ +∞. It is shown that, if the condition fails, the solution can blow up (become infinite) in a finite time. A lower bound for the blow-up time is established. In all the cases, the method of energy inequalities is used as based on the application of the Chaplygin comparison theorem, Bihari-type inequalities, and their difference analogues. A numerical experiment is used to illustrate the theoretical results and verify two-sided blow-up time estimates.

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Authors and Affiliations

  1. Institute of Mathematics, National Academy of Sciences of Belarus, ul. Surganova 11, Minsk, 220072, Belarus

    P. P. Matus

  2. John Paul II Catholic University of Lublin, Al. Raclawickie 14, 20-950, Lublin, Poland

    P. P. Matus

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  1. P. P. Matus
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Dedicated to Academician A.A. Dorodnicyn on the Occasion of the Centenary of His Birth

Original Russian Text © P.P. Matus, 2010, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2010, Vol. 50, No. 12, pp. 2155–2175.

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Matus, P.P. Well-posedness of difference schemes for semilinear parabolic equations with weak solutions. Comput. Math. and Math. Phys. 50, 2044–2063 (2010). https://doi.org/10.1134/S0965542510120079

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