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Solution of Parabolic Equations by Means of Lyapunov Functionals

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Abstract

We propose a new approach to defining the notion of a solution to linear and nonlinear parabolic equations. The main idea consists in studying connections between solutions to dynamic problems in the variational shape and the properties of the corresponding Lyapunov functionals which are strictly decreasing along the trajectories of the above-mentioned dynamic equations except for the equilibrium points. It turns out that the families of Lyapunov functionals constructed by T. I. Zelenyak enable us to propose a new approach to defining solutions to both linear and nonlinear parabolic problems. All results are given in the case of smooth solutions. Note that the Lyapunov functionals can be used for studying solutions with unbounded gradients.

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References

  1. Ladyzhenskaya O. A., Solonnikov V. A., and Ural'tseva N. N., Linear and Quasilinear Equations of Parabolic Type [in Russian], Nauka, Moscow (1967).

    Google Scholar 

  2. Kruzhkov S. N., “Nonlinear parabolic equations in two independent variables,” Trudy Moskov. Mat. Obshch., 16, 329–346 (1967).

    Google Scholar 

  3. Kruzhkov S. N., “Quasilinear and parabolic equations and systems in two independent variables,” Trudy Sem. Petrovsk., No. 5, 217–272 (1979).

  4. Oleinik O. A. and Kruzhkov S. N., “Second-order quasilinear parabolic equations in several independent variables,” Uspekhi Mat. Nauk, 16, No.5, 116–155 (1961).

    Google Scholar 

  5. Krylov N. V., Nonlinear Elliptic and Parabolic Equations of Second Order [in Russian], Nauka, Moscow (1985).

    Google Scholar 

  6. Zelenyak T. I., “Qualitative properties of solutions to quasilinear mixed problems for parabolic type equations,” Mat. Sb., 104, No.3, 486–510 (1977).

    Google Scholar 

  7. Lavrent'ev Jr. M. M., “Estimates of the highest-order derivatives in some nonlinear parabolic equations,” Siberian Math. J., 32, No.1, 72–81 (1991).

    Article  Google Scholar 

  8. Tonelli L., Fondamenti di calcolo della variazioni. Vol. 2, Eds. Zanichelli, Bologna (1923).

  9. Nagumo M., “Uber die gleichmassige Summierbarkeit und ihre Anwendung auf ein Variationsproblema,” Japan J. Math., 6, 173–182 (1929).

    Google Scholar 

  10. Bernstein S. N., Collected Works. Vol. 3: Differential Equations, Variational Calculus, and Geometry [in Russian], Izdat. Akad. Nauk SSSR, Moscow, 1960, pp. 191–242.

    Google Scholar 

  11. Zelenyak T. I., Lavrentiev Jr. M. M., and Vishnevskii M. P., Qualitative Theory of Parabolic Equations, VSP, Utrecht, The Netherlands (1997).

    Google Scholar 

  12. Belonosov V. S. and Zelenyak T. I., Nonlocal Problems in the Theory of Quasilinear Parabolic Equations [in Russian], Novosibirsk Univ., Novosibirsk (1975).

    Google Scholar 

  13. Zelenyak T. I., “On stabilization of solutions to boundary value problems for a second-order parabolic equation with one space variable,” Differentsial'nye Uravneniya, 4, No.1, 34–45 (1968).

    Google Scholar 

  14. Lavrentiev Jr. M. M., Broadbridge P., and Belov V., “Boundary value problems for strongly degenerate parabolic equations,” Comm. Partial Differential Equations, 22, No.1–2, 17–38 (1997).

    Google Scholar 

  15. Filippov A. F., “Conditions for existence of solutions to a quasilinear parabolic equation,” Dokl. Akad. Nauk SSSR, 141, No.3, 568–570 (1961).

    Google Scholar 

  16. Dlotko T., “Examples of parabolic problems with blowing-up derivatives,” J. Math. Anal. Appl., 154, 226–237 (1991).

    Article  Google Scholar 

  17. Samarskii A. A., Galaktionov V. A., Kurdyumov S. P., and Mikhailov A. P., Blow-Up in Quasilinear Parabolic Equations, de Gruyter, Berlin (1995). (de Gruyter Exposition in Mathematics; V. 19).

    Google Scholar 

  18. Samarskii A. A. and Mikhailov A. P., Mathematical Modeling: Ideas, Methods, Examples [in Russian], Nauka, Moscow (1997).

    Google Scholar 

  19. Fila M. and Lieberman G., “Derivative blow-up and beyond for quasilinear parabolic equations,” Differential Integral Equations, 7, No.3/4, 811–822 (1994).

    Google Scholar 

  20. Angenent S. and Fila M., “Interior gradient blow-up in a semilinear parabolic equation,” Differential Integral Equations, 9, No.5, 865–877 (1996).

    Google Scholar 

  21. Bertsch M. and Dal Passo R., “A parabolic equation with a mean-curvature type operator,” in: Nonlinear Diffusion Equations and Their Equilibrium States. 3. Eds. N. G. Lloyd et al., Birkhauser, Boston; Basel; Berlin, 1992, pp. 89–97.

    Google Scholar 

  22. Bertsch M., Dal Passo R., and Franchi B., “A degenerate parabolic equation in noncylindrical domains,” Math. Ann., 294, No.3, 551–578 (1992).

    Article  Google Scholar 

  23. Bertsch M. and Dal Passo R. “Hyperbolic phenomena in a strongly degenerate parabolic equation,” Arch. Rational Mech. Anal., 117, No.4, 349–387 (1992).

    Article  Google Scholar 

  24. Barenblatt G. I., Bertsch M., Dal Passo R., Prostokishin V. M., and Ughi M., “A mathematical model of turbulent heat and mass transfer in stably stratified shear flow,” J. Fluid Mech., 253, 34–45 (1993).

    Google Scholar 

  25. Lavrentiev Jr. M. M., “Gradient Blowing-Up” Solutions to Parabolic Problems: Examples and Available Solvability Results [Preprint, No. 3.99], School of Math. and Applied Stat. University of Wollongong, Wollongong, Australia (1999).

    Google Scholar 

  26. Vishnevskii M. P., Zelenyak T. I., and Lavrent'ev Jr. M. M., “Behavior of solutions to nonlinear parabolic equations at large time,” Siberian Math. J., 36, No.3, 435–453 (1995).

    Article  Google Scholar 

  27. Ball J. M. and Mizel V. J., “Singular minimizers in the calculus of variations,” Bull. Amer. Math. Soc. (N. S.), 11, 143–146 (1984).

    Google Scholar 

  28. Sychev M. A., “To the question of regularity of solutions to variational problems,” Mat. Sb., 183, No.4, 118–142 (1992).

    Google Scholar 

  29. Davie A. M. “Singular minimizers in the calculus of variations in one dimension,” Arch. Rational Mech. Anal., 101, No.2, 161–177 (1988).

    Article  Google Scholar 

  30. Sychev M. A., Characterization of Properties for Weak-Strong Convergence of Integral Functionals in the Terms of Integrands [Preprint, No. 11] [in Russian], Inst. Mat. (Novosibirsk), Novosibirsk (1994).

    Google Scholar 

  31. Lavrent'ev Jr M. M., “Solvability of nonlinear parabolic problems,” Siberian Math. J., 34, No.6, 1110–1116 (1993).

    Article  Google Scholar 

  32. Crandall M. G., Ishii H., and Lions P. L., “User's guide to viscosity solutions of second order partial differential equations,” Bull. Amer. Math. Soc., 27, 1–67 (1992).

    Google Scholar 

  33. Matano H., “Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation,” J. Fac. Sci. Univ. Tokyo, Sect. I A, 29, No.2, 401–441 (1982).

    Google Scholar 

  34. Angenent S., “The zero set of a solution of a parabolic equation,” J. Reine Angew. Math., 390, 79–96 (1988).

    Google Scholar 

  35. Angenent S., “Nodal properties of solutions of parabolic equations,” Rocky Mountain J. Math., 21, No.2, 585–592 (1991).

    MathSciNet  Google Scholar 

  36. Brunovsky P., Polacik P., and Sandstede B., “Convergence in general periodic parabolic equations in one space dimension,” Nonlinear Anal., 18, No.3, 209–215 (1992).

    Article  Google Scholar 

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    M. M. Lavrent'ev Jr.

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  1. M. M. Lavrent'ev Jr.
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Original Russian Text Copyright © 2005 Lavrent'ev Jr. M. M.

The author was supported by the Russian Foundation for Basic Research (Grants 00-07-90343; 00-15-99092).

In memory of Tadei Ivanovich Zelenyak.

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Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 5, pp. 1085–1099, September–October, 2005.

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Lavrent'ev, M.M. Solution of Parabolic Equations by Means of Lyapunov Functionals. Sib Math J 46, 867–878 (2005). https://doi.org/10.1007/s11202-005-0085-z

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  • DOI: https://doi.org/10.1007/s11202-005-0085-z

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