Summary
An interfacial phenomenon for a class of the solutions of a nonlinear forward-backward parabolic equation in R × (0,T) is investigated. In general, short time-period of interfaces is considered. This inner analysis allows to construct on some time interval a solution of the Cauchy problem for certain initial data.
Article PDF
Similar content being viewed by others
References
J. L.Lions,Quelques méthodes de resolution des problèmes aux limites non linéaires, Paris (1968).
V. N.Geebenev,On a solvability of non-standard problems for nonlinear parabolic equations with variable time direction, Ph.D. Thesis, Novosibirsk State University (1987).
J. L. Vazquez,The interfaces of one-dimensional flows in porous media, Trans. Amer. Math. Soc.,285 (1984), pp. 717–737.
K. Hollig,Existence of infinitely many solutions for a forward-backward heat equation, Trans. Amer. Math. Soc.,278 (1983), pp. 299–316.
K.Hollig - J. A.Nohel,A diffusion equation with a nonmonotone constitutive function, Nonlinear Partial Differential Equations, Dordrecht (1983), pp. 409–422.
R. R. Akhmerov,On structure of a set of solutions of Dirichlet boundary value problem for stationary one-dimensional forward-backward parabolic equation, Nonlinear Analysis, TMA,11 (1987), pp. 1303–1316.
R. J. DiPerna,Measure-valued solutions to conservation laws, Arch. Ration. Mech. Anal.,88 (1985), No. 3, p. 1985.
M. Slemrod,Measure-valued solutions to a backward-forward heat equation: a conference report, inNonlinear Evolution Equations that Change Type, The IMA Volumes in Math. and Appl., No. 27, pp. 232–242, Springer-Verlag, New York (1990).
P. I. Plotnikov,Forward-backward parabolic equations and hysteresis, Russian Math. Dokl.,330 (1993), pp. 691–694.
Alan V. Lair,Uniqueness for a forward backward diffusion equation, Trans. Amer. Math. Soc.,291 (1985), pp. 311–317.
Alan V. Lair,Uniqueness for a forward backward diffusion equation with smooth constitutive function, Applicable Anal.,29 (1988), pp. 177–189.
R. R.Akhmerov,On periodic travelling waves of equations with viscosity coefficient of variable sign, Nonlinear Analysis, TMA,13 (19)89, pp. 803–817.
J. W. Cahn,On the spinodal decomposition, Acta Metall.,9 (1961), pp. 795–801.
C. M. Elliott,The Cahn-Hilliard model for the kinetics of phase separation, inMathematical Models for Phase Change Problems, in Int. Series of Numer. Math., No. 88, pp. 35–73, Birkhauser Verlag, Basel (1989).
N. N. Yanenko -V. A. Novikov,On certain model of a fluid with viscosity of variable sign, Chisl. Met. Mekh. Sploshn. Sredy,11 (1973), pp. 142–147, Novosibirsk.
O. A. Ladyzewnskaja -V. A. Solonikov -N. N. Uralceva,Linear and Quasilinear Equations of Parabolic Type, Trans. Math. Monogr.,23, A.M.S., Rhode Island (1986).
B. F. Knerr,The porous medium equation in one dimension, Trans. Amer. Math. Soc.,234 (1977), pp. 381–403.
A. S. Kalashnikov,On the occurence of singularities in the solutions of the equation of nonstationary filtration, Z. Vycisl. Mat. i Mat. Fiz.,7 (1967), pp. 440–444.
A. Friedman,One dimensional Stefan problems with nonmonotone free boundary, Trans. amer. Math. Soc.,133 (1968), pp. 89–114.
M. Bertsch -D. Hilhorst,The interface between regions where u > 0and u < 0in the porous medium equation, Preprint Univer. de Paris-Sud Math.,10 (1989), pp. 1–25.
A. S. Kalashnikov,The effect of absorption in a medium in which thermal conductivity depends on temperature, USSR comp. Math. and Math. Math. Phiz.,16∶3 (1976), pp. 141–149.
D. Aronson -M. G. Crandall -L. A. Paltrier,Stabilization of solutions of a degenerate nonlinear diffusion problem, Nonlinear analysis, TMA,6 (1982), pp. 1001–1022.
D.Aronson - L. A.Caffarelli - J. L.Vazquez,Interfaces with a corner point in one-dimensional porous medium flow, Commun. Pure Appl. Math.,38, pp. 375–404.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Grebenev, V.N. Interfacial phenomenon for one-dimensional equation of forward-backward parabolic type. Annali di Matematica pura ed applicata 171, 379–394 (1996). https://doi.org/10.1007/BF01759392
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01759392