Abstract
In this paper, we study a class of doubly nonlinear parabolic PDEs, where, in addition to some weak nonlinearities, also some mild nonlinearities of porous media type are allowed inside the time derivative. In order to formulate the equations as dynamical systems, some existence and uniqueness results are proved. Then the existence of a compact attractor is shown for a class of nonlinear PDEs that include doubly nonlinear porous medium-type equations. Under stronger smoothness assumptions on the nonlinearities, the finiteness of the fractal dimension of the attractor is also obtained.
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References
Alt, H. W., and Luckhaus, S. (1983). Quasilinear elliptic-parabolic equations.Math. Z. 183, 311–341.
Aronson, D. G. (1981). Nonlinear diffusion problems. In Fasano and Primiciero (eds.),Free Boundary Problems: Theory and Applications, Vol. 1, Pitman Research Notes in Mathematics.
Aronson, D. G., and Peletier, L. A. (1981). Large time behaviour of solutions of the porous medium equation in bounded domain.J. Diff. Eq. 39, 378–412.
Barbu, V. (1976).Semigroups and Differential Equations in Banach Spaces, Noordhooff.
Benilan, P. (1972).Equation d'évolution dans un espace de Banach quelconque et applications, Thèse, Université Paris XI, Orsay.
Benilan, P. (1974). Opérateursm-accrétifs hémicontinus dans un espace de Banach quelconque.C.R. Acad. Sci. Paris Ser. A 278, 1029–1032.
Benilan, P. (1977). Existence de solutions fortes pour l'équation des milieux poreux.C.R. Acad. Sci. Paris Ser. A 285, 1029–1031.
Benilan, P., Brezis, H., and Crandall, M. G. (1975). A semilinear equation inL 1ℝN).Ann. Scula Norm. Sup. Pisa Cl. Sci. (4) 2, 523–555.
Benilan, P., Crandall, M. G., and Pierre, M. (1984). Solution of the porous medium equation in ℝN, under optimal condition.Ind. Univ. J. 33, No. 1.
Blanchard, D., and Francfort, G. (1988). Study of a doubly nonlinear heat equation with no growth assumptions on the parabolic term.Siam J. Num. Anal. 19, No. 5.
Brezis, H. (1973).Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland, Amsterdam.
Brezis, H. (1984).Analyse Fonctionnelle: Théorie et Applications, Masson, Paris.
Brezis, H., and Crandall, M. G. (1979). Uniqueness of solutions of the initial value problem foru t,-Δϕ(u)=0.J. Math. Pure Appl. 58, 153–163.
Brezis, H., and Friedman, A. (1983). Nonlinear parabolic equation involving measures as initial data.J. Math. Pure Appl. 62, 73–97.
Brezis, H., and Pazy, A. (1970). Accretive sets and differential equations in Banach spaces.Israel J. Math. 8, 367–383.
Constantin, P., Foias, C, and Temam, R. (1985). Attractors representing turbulent flows.AMS Memoirs 53, No. 314.
Crandall, M. G. (1975). An introduction to evolution governed by accretive operators. InDynamical Systems. Proceeding of an International Symposium, Brown University, 1974, Academic Press, New York-San Francisco-London.
Damlamian, A. (1977). Some results on the multiphase Stefan problem.Par. Diff. Eqs. 2, 1017–1044.
Eden, A. (1989). On Burgers' original mathematical model of turbulence. Preprint 8905, Inst. Appl. Math. Sci. Comp., Indiana University, Bloomington.
Ekeland, I., and Temam, R. (1976).Analyse Convexe et Problèmes Variationnels, Dunod, Paris.
Foias, C., and Temam, R. (1977). Structure of the set of stationary solutions of the NavierStokes equations.Comm. Pure Appl. Math. 30, 149–164.
Friedman, A. (1969).Parabolic Differential Equations, Holt, Rienhart and Winston, New York.
Friedman, A. (1982).Variational Principles and Free Boundary Problems, Wiley, New York.
Friedman, A., and Kamin, S. (1980). The asymptotic behaviour of a gas inn-dimensional porous medium.Trans. Am. Math. Soc. 262, 551–563.
Ghidaglia, J. M., and Héron, B. (1987). Dimension of the attractor associated with Ginsburg-Landau partial differential equation.Physica 28D, 282–304.
Grange, O., and Mignot, E. (1972). Sur la résolution d'une inéquation parabolique nonlinéaire.J. Funct. Anal. 11, 77–92.
Gurney, W. S. C., and Nisbet, R. M. (1975). The regulation of inhomogeneous populations.J. Theoret. Biol. 52, 441–557.
Gurtin, M. E., and Mac Camy, R. C. (1977). On the diffusion of biological populations.Math. Bio. Sci 33, 35–49.
Jolly, M. S. (1989). Explicit construction of an inertial manifold for a reaction diffusion equation.J. Diff. Eqs. 78, 220–261.
Kamin, S. (1976). Similar solutions and the asymptotics of filtration equations.Arch. Rat. Mech. Anal. 60, 171–183.
Ladyzenskaja, O. A., and Ural'tseva, N. N. (1985). On the solvability of the first boundary value problem for quasilinear elliptic and parabolic equations in the presence of singularity,Soviet Math. Dokl.31, No. 2.
Ladyzenskaja, O. A., Solonnikov, V. A., and Ural'tseva, N. N. (1968).Linear and QuasiLinear Equations of Parabolic Type, AMS Transl. Monogr. 23, Providence, R.I.
Lions, J. L. (1969).Quelque Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Paris Dunod. Mallet-Paret, J., and Sell, G. R. (1987). Inertial manifolds for reaction diffusion equations in higher space dimension, IMA preprint No. 331.
Marion, M. (1987). Attractors for reaction-diffusion equations: Existence and estimate of their dimension.Appl. Anal. 25, 101–147.
Massey, F. J. (1977). Semilinear parabolic equations withL 1 initial data.Ind. Univ. J. 26, 399–411.
Nicolaenko, B., Scheurer, B., and Temam, R. (1990). Some global dynamical properties of a class of pattern formation equations.Comm. Part. Diff. Eq. (in press).
Rakotoson, J. M. (1986). Time behaviour of solutions of parabolic problems.Appl. Anal. 21, 13–29.
Raviart, P. A. (1970). Sur la résolution de certaines équations paraboliques nonlinéaires.J. Funct. Anal. 5, 299–328.
Sabinina, E. S. (1961). On the Cauchy problem for the equation of non-stationary gas filtration in several space variables.Dokl. Akad. Nauk SSSR,136, 1034–1037.
Schatzman, M. (1984). Stationary solutions and asymptotic behavior of a quasilinear parabolic equation.Ind. Univ. J. 33 (1), 1–29.
Sermange, M., and Temam, R. (1983). Some mathematical questions related to MHD equations.Comm. Pure Appl. Math. 36, 635–664.
Temam, R. (1988).Infinite Dimensional Dynamical Systems in Mechanics and Physics, Appl. Math. Sci.68, Springer-Verlag, New York.
Temam, R. (1984).Navier-Stokes Equations: Theory and Numerical Analysis, North-Holland, Amsterdam.
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Eden, A., Michaux, B. & Rakotoson, J.M. Doubly nonlinear parabolic-type equations as dynamical systems. J Dyn Diff Equat 3, 87–131 (1991). https://doi.org/10.1007/BF01049490
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DOI: https://doi.org/10.1007/BF01049490