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Nonlinear problems of the thermal conductivity equation

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Journal of Applied Mechanics and Technical Physics Aims and scope

Abstract

The asymptotic behavior of solutions of parabolic equations at infinite times has been investigated for various cases [1–6]. Two initial boundary-value problems are considered in this paper. The solution of the thermal conductivity equation with a nonlinear right-hand side is found, including also nonlinear boundary conditions. It is shown that the solution of the corresponding problem tends either to a stable, steady-state solution, or to a periodic solution, depending on the initial values of the functions and constants appearing in the conditions of the problem. Other papers [7, 8] are devoted to finding the periodic solutions of these two problems encountered in hydrodynamics (diffusion, underground hydrodynamics), and to studying the asymptotic behavior of the corresponding initial boundary problems.

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Literature cited

  1. M. I. Vishik and L. A. Lyusternik, “Stabilization of solutions of some differential equations in Hubert space,” Dokl. Akad. Nauk SSSR,111, No. 1 (1956).

  2. M. I. Vishik and L. A. Lyusternik, “Stabilization of solutions of parabolic equations,” Dokl. Akad. Nauk SSSR,111, No. 2 (1956).

  3. Yu. P. Gor'kov, “Periodic solutions of parabolic equations,” Diff. Urav.,2, No. 7 (1966).

  4. Yu. P. Gor'kov, “Behavior of solutions of boundary value problems for second-order quasilinear parabolic equations at t → ∞,” Diff. Urav.,6, No. 6 (1970).

  5. T. I. Zelenyak, “Stabilization of solutions of boundary value problems for second-order parabolic equations with one space variable,” Diff. Urav.,4, No. 1 (1968).

  6. E. N. Rudenko, “Asymptotic stability of solutions of parabolic equations,” Diff. Urav.,6, No. 1 (1970).

  7. N. N. Kochina, “Varying levels of ground water at irrigation,” Zh. Prikl. Mekh. i Tekh. Fiz.,12, No. 4 (1971).

  8. N. N. Kochina, “Asymptotic behavior of solutions of some nonlinear problems in hydrodynamics,” Prikl. Matern, i Mekhan.,35, No. 6 (1971).

  9. Yu. S. Kolesov, “Periodic solutions of a class of differential equations with nonlinear hysteresis,” Dokl. Akad. Nauk SSSR,176, No. 6 (1967).

  10. Yu. S. Kolesov, “Periodic solutions of Rayleigh systems with parameter distributions,” Matem. sb.,83, No. 3 (1970).

  11. A. Ya. Gokhshtein, “Stability of stationary states of electrolytic systems,” Dokl. Akad. Nauk SSSR,149, No. 4 (1963).

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Authors and Affiliations

  1. Moscow

    N. N. Kochina

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  1. N. N. Kochina
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Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 123–128, May–June, 1972.

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Kochina, N.N. Nonlinear problems of the thermal conductivity equation. J Appl Mech Tech Phys 13, 367–371 (1972). https://doi.org/10.1007/BF00850429

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  • DOI: https://doi.org/10.1007/BF00850429

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