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Stability with respect to the initial data and monotonicity of an implicit difference scheme for a homogeneous porous medium equation with a quadratic nonlinearity

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Abstract

We study the stability and monotonicity of a conservative difference scheme approximating an initial-boundary value problem for a porous medium equation with a quadratic nonlinearity under certain conditions imposed only on the input data of the problem. We prove a grid analog of the Bihari lemma, which is used to obtain a priori estimates for higher derivatives; these estimates are needed both in the proof of the continuous dependence of the solution on small perturbations in the input data and for the analysis of monotonicity in the nonlinear case. We show that, regardless of the smoothness of the initial condition, the higher derivatives can become infinite in finite critical time. We give an example in which there arises a runningwave solution, which justifies the theoretical conclusions.

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References

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Authors and Affiliations

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

    P. P. Matus

  2. Catholic University, Lublin, Poland

    P. P. Matus

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  1. P. P. Matus
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Original Russian Text © P.P. Matus, 2010, published in Differentsial’nye Uravneniya, 2010, Vol. 46, No. 7, pp. 1011–1021.

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Matus, P.P. Stability with respect to the initial data and monotonicity of an implicit difference scheme for a homogeneous porous medium equation with a quadratic nonlinearity. Diff Equat 46, 1019–1029 (2010). https://doi.org/10.1134/S0012266110070098

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  • DOI: https://doi.org/10.1134/S0012266110070098

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