Abstract
This paper is concerned with the asymptotic behavior of weak solutions to a multi-phase Stefan problem for a quasi-linear heat equation of the form ρ(v) t - Δv=ƒ in several space variables, with Dirichlet-Neumann boundary condition on the fixed boundary. We shall discuss the asymptotic convergence of the enthalpy and temperature inL 2(Ω) andH 1(Ω), respectively, when the prescribed boundary data asymptotically converge in some sense. Our approach to the investigation of the asymptotic convergence of solutions is based on the theory of nonlinear evolution equations governed by time-dependent subdifferential operators in Hilbert spaces. The results obtained in this paper improve on those established so far.
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References
H. Attouch, Ph. Bénilan, A. Damlamian and C. Picard, Équation d’évolution avec condition unilatérale. C. R. Acad. Sci. Paris,279 (1974), 607–609.
H. Attouch and A. Damlamian, Problèmes d’évolution dans les Hilbert et applications. J. Math. Pures Appl.,54 (1975), 53–74.
M. Biroli, Sur les inéquations paraboliques avec convexe dépendant du temps, solution forte et solution faible. Riv. Mat. Univ. Parma,3 (1974), 33–72.
H. Brézis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert. North-Holland, Amsterdam-London, 1973.
J. R. Cannon and M. Primicerio, A two phase Stefan problem with temperature boundary conditions. Ann. Mat. Pura Appl.,88 (1971), 177–191.
J. R. Cannon and M. Primicerio, A two phase Stefan problem with flux boundary conditions. Ann. Mat. Pura Appl.,88 (1971), 193–205.
A. Damlamian, Problèmes aux limites non linéaires du type du problème de Stefan et inéquations variationnelles d’évolution. Thèse, Univ. Paris VI, 1976.
A. Damlamian, Some results on the multi-phase Stefan problem. Comm. Partial Differential Equations,2 (1977), 1017–1044.
A. Friedman, The Stefan problem in several space variables. Trans. Amer. Math. Soc.,133 (1968), 51–87.
H. Furuya, K. Miyashiba and N. Kenmochi, Asymptotic behavior of solutions to a class of nonlinear evolution equations. To appear in J. Differential Equations.
S. L. Kamenomostskaja, On Stefan’s problem. Mat. Sb.,53 (1961), 489–514.
N. Kenmochi, Some nonlinear parabolic variational inequalities. Israel J. Math.,22 (1975), 304–331.
N. Kenmochi, On the quasi-linear heat equation with time-dependent obstacles. Nonlinear Anal.,5 (1981), 71–80.
N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications. Bull. Fac. Ed., Chiba Univ.,30 (1981), 1–87.
N. Kenmochi and M. Kubo, Periodic solutions to a class of nonlinear variational inequalities with time-dependent constraints. Preprint.
O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural’ceva, Linear and Quasi-Linear Equations of Parabolic Type. Transl. Math. Monogr. 23, Amer. Math. Soc., Providence R. I., 1968.
O. A. Oleinik, On a method of solving of the general Stefan problem. Soviet Math. Dokl.,1 (1960), 1350–1354.
M. Ôtani, Nonmonotone perturbations for nonlinear parabolic equations associated with sub-differential operators, Cauchy problems. J. Differential Equations,46 (1982), 268–299.
L. I. Rubinstein, The Stefan Problem. Transl. Math. Monogr. 27, Amer. Math. Soc., Providence R. I., 1971.
Y. Yamada, On evolution equations generated by subdifferentials. J. Fac. Sci. Univ. Tokyo Sect. IA,23 (1976), 491–515.
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Damlamian, A., Kenmochi, N. Asymptotic behavior of solutions to a multi-phase Stefan problem. Japan J. Appl. Math. 3, 15–36 (1986). https://doi.org/10.1007/BF03167089
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DOI: https://doi.org/10.1007/BF03167089