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The free boundary problem without initial condition

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Abstract

We consider an analog of the problem of the impact of a viscoplastic rod on the wall under a nonlinear boundary condition. We investigate the behavior of the free boundary on a given time interval and as t→∞. We obtained a priori estimates of Hölder norms and proved the theorems of uniqueness and existence of the solution.

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References

  1. G. I. Barenblatt and A. Yu. Ishlinskii, “On the impact of a viscoelastic rod on the rigid obstacle,” Prikl. Mat. Mekh., 26, No. 3, 497–502 (1962).

    Google Scholar 

  2. A. Fasano and M. Primiecerio, “Viscoplastic impact of a rod on a wall,” Boll. Un. Mat. Ital., 4(VII), No. 3, 531–555 (1975).

    Google Scholar 

  3. A. M. Il’in, A. S. Kalashnikov, and O. A. Oleinik, “Linear equations of the second order of the parabolic type,” Uspekhi Mat. Nauk, 12, No. 3, 3–141 (1962).

    Google Scholar 

  4. S. N. Kruzhkov, “On some problems with unknown boundary for the heat equation,” Prikl. Mat. Mekh., 31, No. 6, 1009–1020 (1967).

    MathSciNet  Google Scholar 

  5. S. N. Kruzhkov and S. Yakubov, “On the solvability of a class of problems with unknown boundary for the heat equation and on the behavior of solutions under the unlimited increase in time,” Continuum Dynamics [in Russian], Issue 36, Inst. of Hydrodynamics of SD AS SSSR, Novosibirsk, 1978, pp. 46–70.

  6. S. N. Kruzhkov, “Nonlinear parabolic equations with two independent variables,” Trudy Mosk. Mat. Obshch., 16, 329–346 (1967).

    MATH  Google Scholar 

  7. L. I. Rubinstein, The Stefan Problem, Amer. Math. Soc., Providence, RI, 1994.

    Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics and Informational Technologies of the AS of RUz, 29, Durmon iyuli Str., Tashkent, 100125, Uzbekistan

    Jozil O. Takhirov & Rasul N. Turaev

Authors
  1. Jozil O. Takhirov
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  2. Rasul N. Turaev
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Corresponding author

Correspondence to Jozil O. Takhirov.

Additional information

Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 9, No. 2, pp. 259–277, April–May, 2012.

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Takhirov, J.O., Turaev, R.N. The free boundary problem without initial condition. J Math Sci 187, 86–100 (2012). https://doi.org/10.1007/s10958-012-1052-4

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  • DOI: https://doi.org/10.1007/s10958-012-1052-4

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