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Behavior of solutions to nonlinear parabolic equations at large time

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References

  1. A. N. Kolmogorov, I. G. Petrovskii, and N. S. Piskunov, “Study of the equation of diffusion combined with increase of substance and its application to a certain biological problem,” Byulleten' Moskov. Univ. Sektsiya AI, No. 6, 1–26 (1937).

    Google Scholar 

  2. A. K. Gushchin and V. P. Mikhailov, “On stabilization of a solution to the Cauchy problem for a parabolic equation,” Differentsial'nye Uravneniya,7, No. 2, 297–311 (1971).

    Google Scholar 

  3. S. D. Ivasishen and S. D. èidel'man, “Studying Green's matrix for a homogeneous parabolic boundary value problem,” Trudy Moskov. Mat. Obshch.,23, 179–234 (1970).

    Google Scholar 

  4. T. I. Zelenyak and M. G. Slin'ko, “The dynamics of catalytic systems. I,” Kinetika i Kataliz,18, No. 5, 1235–1248 (1977).

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  8. H. Matano, “Convergence of solutions of one-dimensional semilinear parabolic equations,” J. Math. Kyoto Univ.,18, No. 2, 221–227 (1978).

    Google Scholar 

  9. H. Matano, “Asymptotic behavior and stability of solutions of semilinear diffusion equations,” Publ. Res. Inst. Math. Sci.,15, No. 2, 401–454 (1979).

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  12. V. S. Belonosov, “Estimates for solutions to parabolic systems in weighted Hölder classes and some of their applications,” Mat. Sb.,110, No. 2, 163–188 (1979).

    Google Scholar 

  13. T. I. Zelenyak, Qualitative Theory of Boundary Value Problems for Second-Order Quasilinear Equations of Parabolic Type [in Russian], Novosibirsk Univ., Novosibirsk (1972).

    Google Scholar 

  14. M. M. Lavrent'ev Jr., “A priori smoothness for solutions to some equations of variable type,” Mat. Model.,2, No. 9, 145–153 (1990).

    Google Scholar 

  15. V. S. Belonosov, M. P. Vishnevskii, T. I. Zelenyak, and M. M. Lavrent'ev Jr., On Qualitative Properties of Solutions to Parabolic Equations [Preprint, No. 466] [in Russian], Vychisl. Tsentr. (Novosibirsk), Novosibirsk (1983).

    Google Scholar 

  16. M. P. Vishnevskii, “Attraction domains of a stable stationary solution to a parabolic equation,” Dinamika Sploshn. Sredy (Novosibirsk), Novosibirsk, No. 67, 3–20 (1984).

    Google Scholar 

  17. J. M. Ball and V. J. Mizel, “Singular minimizers in the calculus of variations,” Bull. Amer. Math. Soc. (N. S.), No. 11, 143–146 (1984).

    Google Scholar 

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

    Google Scholar 

  19. M. W. Hirsch, “Stability and convergence in strongly monotone dynamical systems,” J. Reine Angew. Math.,383, 1–53 (1988).

    Google Scholar 

  20. P. Poláčik, “Convergence in smooth strongly monotone flows defined by semilinear parabolic equations,” J. Differential Equations,79, No. 1, 89–110 (1989).

    Article  Google Scholar 

  21. M. P. Vishnevskii, “On stabilization of solutions to weakly-coupled cooperation-diffusion parabolic systems,” Mat. Sb.,183, No. 10, 45–62 (1992).

    Google Scholar 

  22. P. Poláčik and K. Rybakovský, Nonconvergent Bounded Trajectories in Semilinear Heat Equations, [Preprint], Universita degli Studi di Trieste (1994).

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

    Google Scholar 

  24. M. P. Vishnevskii, “Bounded solutions to nonlinear parabolic equations,” in: Partial Differential Equations [in Russian], Inst. Mat. (Novosibirsk), Novosibirsk, 1987, pp. 37–59.

    Google Scholar 

  25. O. A. Ladyzhenskaya and N. N. Ural'tseva, “A survey of results on the solvability of boundary value problems for quasilinear uniformly elliptic and parabolic equations of second order with unbounded singularities,” Uspekhi Mat. Nauk,41, No. 5, 59–83 (1986).

    Google Scholar 

  26. M. P. Vishnevskii, “On some qualitative properties of periodic solutions to quasilinear parabolic equations,” Dinamika Sploshn. Sredy (Novosibirsk),64, 11–23 (1984).

    Google Scholar 

  27. O. G. Provorova, “To the question of behavior at large time of solutions to parabolic equations,” Differentsial'nye Uravneniya,5, No. 1, 108–114 (1969).

    Google Scholar 

  28. P. Brunovský and B. Fiedler, “Connecting orbits in scalar reaction diffusion equations. I,” Dynam. Report.,1, 57–89 (1988).

    Google Scholar 

  29. P. Brunovský and B. Fiedler, “Connecting orbits in scalar reaction diffusion equations. II: The complete solution,” J. Differential Equations,81, No. 1, 106–135 (1989).

    Article  Google Scholar 

  30. T. I. Zelenyak, V. A. Novikov, and N. N. Yanenko, “On properties of a solution of nonlinear equations of variable type,” in: Numerical Methods of Continuum Mechanics [in Russian], Novosibirsk, 1974,5, No. 4, 35–47.

    Google Scholar 

  31. N. A. Lar'kin, N. N. Yanenko, and V. A. Novikov, Nonlinear Equations of Variable Type [in Russian], Nauka, Novosibirsk (1983).

    Google Scholar 

  32. C. M. Elliott and D. A. French, “A nonconforming finite-element method for the two-dimensional Cahn-Hilliard equation,” SIAM J. Numer. Anal.,26, No. 4, 884–903 (1989).

    Article  Google Scholar 

  33. J. W. Cahn and J. E. Hilliard, “Free energy of a nonuniform system. I. Interfacial free energy,” J. Chem. Phys.,28, No. 2, 258–267 (1958).

    Article  Google Scholar 

  34. N. D. Alikakos, P. W. Bates, and G. Fusco, “Slow motion for the Cahn-Hilliard equation in one space dimension,” J. Differential Equations,90, No. 1, 81–135 (1991).

    Article  Google Scholar 

  35. P. W. Bates and S. Zheng, “Inertial manifolds and inertial sets for the phase-field equations,” J. Dynamics Differential Equations,4, No. 2, 375–398 (1992).

    Article  Google Scholar 

  36. D. Furihata, T. Onda, and M. Mori, “A finite difference scheme for the Cahn-Hilliard equation based on the Lyapunov functional,” GAKUTO Intern. Series, Math. Sci. Appl.,2, 347–358 (1993).

    Google Scholar 

  37. P. Brunovský, P. Poláčik and B. Sandestede, “Convergence in general periodic parabolic equations in one space dimension,” Nonlinear Anal.,18, No. 3, 209–215 (1992).

    Article  Google Scholar 

  38. B. Fiedler and J. Mallet-Paret, “A Poincaré-Bendixson theorem for scalar reaction diffusion equations,” Arch. Rational Mech. Anal.,107, No. 4, 325–345 (1989).

    Google Scholar 

  39. T. A. Akramov and T. I. Zelenyak, “On the number of stationary solutions and attraction domains for stationary quasilinear parabolic equations,” in: Mathematical Problems of Chemistry. I [in Russian], Inst. Kataliza Sibirsk. Otdel. Akad. Nauk SSSR, Novosibirsk, 1975, pp. 144–150.

    Google Scholar 

  40. M. P. Vishnevskii, “On nonlocal behavior of solutions to quasilinear mixed problems for equations of parabolic type,” Mat. Model.,2, No. 4, 67–77 (1989).

    Google Scholar 

  41. A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov, Peaking Modes in Problems for Quasilinear Parabolic Equations [in Russian], Nauka, Moscow (1987).

    Google Scholar 

  42. B. Fiedler and C. Rocha, Heteroclinic Orbits of Semilinear Parabolic Equations, [Preprint, No. 14A94], Freie UniversitÄt Berlin, Berlin (1994).

    Google Scholar 

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Translated from Sibirskii Matematicheskii Zhurnal, Vol. 36, No. 3, pp. 510–530, May–June, 1995.

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Vishnevskii, M.P., Zelenyak, T.I. & Lavrent'ev, M.M. Behavior of solutions to nonlinear parabolic equations at large time. Sib Math J 36, 435–453 (1995). https://doi.org/10.1007/BF02109832

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