References
T. I. Zelenyak, “On stabilization of solutions to boundary value problems for a parabolic second order equation with one space variable,” Differential'nye Uravneniya,4, 34–45 (1968).
T. I. Zelenyak, “On qualitative properties of solutions to quasilinear mixed problems for equations of parabolic type,” Mat. Sb.,104, 486–510 (1977).
V. S. Belonosov and T. I. Zelenyak, Nonlocal Problems in the Theory of Quasilinear Parabolic Equations [in Russian], Novosibirsk Univ., Novosibirsk (1972).
H. Matano, “Convergence of solutions of one-dimensional semilinear parabolic equations,” J. Math. Kyoto Univ.,18, 221–227 (1978).
H. Matano, “Nonincrease of the lap-number of a solution for one-dimensional semilinear parabolic equations,” J. Fac. Sci. Univ. Tokyo Sect. IA Math.,29, 401–441 (1982).
J. K. Hale, “Asymptotic behavior and dynamics in infinite dimensions,” in: Nonlinear Differential Equations (Research Notes Math.,132), Pitman, Boston, 1985.
P. L. Lions, “Structure of the set of steady state solutions and asymptotic behavior of semilinear heat equations,” J. Differential Equations,59, 362–386 (1984).
M. P. Vishnevskiî, “Bounded solutions to nonlinear parabolic equations,” in: Partial Differential Equations [in Russian], Inst. Mat., Novosibirsk, 1987, pp. 37–59.
M. P. Vishnevskiî, “Asymptotic behavior of solutions to mixed problems for quasilinear parabolic equations,” in: Boundary Value Problems for Partial Differential Equations [in Russian], Inst. Mat., Novosibirsk, 1988, pp. 65–82.
M. P. Vishnevskiî, “On the behavior at large time of solutions to quasilinear parabolic equations with many space variables,” Dinamika Sploshn. Sredy (Novosibirsk),85, 34–41 (1988).
M. P. Vishnevskiî, “Periodic solutions to autonomous parabolic equations,” Dinamika Sploshn. Sredy (Novosibirsk),98, 98–106 (1990).
M. W. Hirsch, “Stability and convergence in strongly monotone dynamical systems,” J. Reine Angew. Math.,383, 1–53 (1988).
P. Polačik, “Convergence in smooth strongly monotone flows defined by semilinear parabolic equations,” J. Differential Equations,79, 89–110 (1989).
N. S. Nadirashvili, “On the dynamics of nonlinear parabolic equations,” Dokl. Akad. Nauk SSSR,309, No. 6, 1302–1305.
A. V. Babin and M. I. Vishik, Attractors for Evolution Equations [in Russian], Nauka, Moscow (1989).
A. V. Babin and M. I. Vishik, “Uniform finite-parameter asymptotics of solutions of nonlinear evolutionary equations,” J. Math. Pures Appl.,68, 399–455 (1989).
O. A. Ladyzhenskaya and N. N. Ural'tseva, “A survey of results on solvability of boundary value problems for uniformly elliptic and parabolic quasilinear second order equations with unbounded singularities,” Uspekhi Mat. Nauk,41, 59–83 (1986).
M. P. Vishnevskiî, “Integral sets of nonlinear parabolic systems,” in: Some Applications of Functional Analysis to Problems of Mathematical Physics [in Russian], Novosibirsk, 1986, pp. 32–56.
M. P. Vishnevskiî, “On nonlocal behavior of solutions to quasilinear mixed problems for equations of parabolic type,” Mat. Model.,2, No. 4, 67–77 (1990).
V. S. Belonosov, “Estimates for solutions to nonlinear parabolic systems in weighted Hölder classes and some their applications,” Mat. Sb.,110, 163–188 (1979).
V. S. Belonosov and M. P. Vishnevskiî, “On stability of stationary solutions to nonlinear parabolic systems,” Mat. Sb.,104, 535–558 (1977).
Yu. S. Kolesov, “Stability of solutions to parabolic equations,” Izv. Akad. Nauk SSSR Ser. Mat.,33, 1356–1372 (1966).
M. H. Protter and H. Wainberger, Maximum Principles in Differential Equations, Springer, Berlin (1984).
P. Hartman, Ordinary Differential Equations [Russian translation], Mir, Moscow (1970).
P. Polačik, “Domain of attractors of equilibria and monotonicity properties of convergence trajectories in parabolic systems admitting strong comparison principle,” J. Reine Angew. Math.,400, 32–56 (1989).
Additional information
Novosibirsk. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 34, No. 4, pp. 50–60, July–August, 1993.
Translated by G. V. Dyatlov
Rights and permissions
About this article
Cite this article
Vishnevskiî, M.P. Monotone solutions to quasilinear parabolic equations. Sib Math J 34, 636–645 (1993). https://doi.org/10.1007/BF00975164
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00975164