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Existence results for doubly nonlinear higher order parabolic equations on unbounded domains. (English) Zbl 0609.35048
We prove the existence of weak (or ”energy”) solutions of the homogeneous Dirichlet initial-boundary value problem for some equations of the form \(\partial (Bu)/\partial t+Au=f\), where A and B are nonlinear monotone operators deriving from convex functionals and the spatial domain is an arbitrary open set of \({\mathbb{R}}^ n\). In particular, our existence theorem applies (for any \(p,q>1\) and any m,n\(\geq 1)\) if A and B are defined by: \[ Au=(-1)^ m\sum _{| \alpha | =m}D^{\alpha}(| D^{\alpha}u| ^{p-1}sgn D^{\alpha}u),\quad (Bu)(x)=| u(x)| ^{q-1}sgn u(x). \] We start from an existence result of O. Grange and F. Mignot [J. Funct. Anal. 11, 77-92 (1972; Zbl 0251.35055)] and follow some methods of H. W. Alt and S. Luckhaus [Math. Z. 183, 311-341 (1983; Zbl 0497.35049)]. In addition, we use Nikol’skiĭ spaces (spaces involving Hölder conditions in the \(L^ p\) metric) to perform a key compactness argument of the proof. We also prove an abstract formula of integration by parts which allows to handle the very weak derivative \(\partial (Bu)/\partial t\) and implies an ”energy estimate for our weak solutions.

MSC:
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
47H05 Monotone operators and generalizations
35D05 Existence of generalized solutions of PDE (MSC2000)
35K65 Degenerate parabolic equations
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References:
[1] Adams, R.A.: Sobolev spaces. New York: Academic Press 1975 · Zbl 0314.46030
[2] Alt, H.W., DiBenedetto, E.: Nonsteady flow of water and oil through inhomogeneous porous media. Ann. Sc. Norm. Super. Pisa Cl. Sci. IV Ser.12, 335-392 (1985) · Zbl 0608.76082
[3] Alt, H.W., Luckhaus, S.: Quasilinear elliptic-parabolic differential equations. Math. Z.183, 311-341 (1983) · Zbl 0508.35046 · doi:10.1007/BF01176474
[4] Atkinson, C., Bouillet, J.E.: Some qualitative properties of solutions of a generalised diffusion equation. Math. Proc. Cam. Philos. Soc.86, 495-510 (1979) · Zbl 0428.35046 · doi:10.1017/S030500410005636X
[5] Aubin, J.-P.: Un théorème de compacité. C.R. Acad. Sci. Paris Ser. I256, 5042-5044 (1963) · Zbl 0195.13002
[6] Bamberger, A.: Étude d’une équation doublement non linéaire. Rapport Interne No. 4 du Centre de Mathématiques Appliquées de l’Ecole Polytechnique, Palaiseau 1-34 (1975)
[7] Bamberger, A.: Étude d’une équation doublement non linéaire. J. Funct. Anal.24, 148-155 (1977) · Zbl 0345.35059 · doi:10.1016/0022-1236(77)90051-9
[8] Barbu, V.: Nonlinear semigroups and differential equations in Banach spaces. Leyden: Noordhoff 1976 · Zbl 0328.47035
[9] Bernis, F.: Qualitative properties for some nonlinear higher order degenerate parabolic equations. IMA Preprint 184, University of Minnesota, 1985. Houston J. Math. (to appear) · Zbl 0682.35009
[10] Bernis, F.: Finite speed of propagation and asymptotic rates for some nonlinear higher order parabolic equations with absorption. IMA Preprint 185, University of Minnesota, 1985. Proc. R. Soc. Edinb.104A, 1-19 (1986) · Zbl 0627.35050
[11] Bouillet, J.E., Atkinson, C.: A generalized diffusion equation: radial symmetries and comparison theorems. J. Math. Anal. Appl.95, 37-68 (1983) · Zbl 0536.35036 · doi:10.1016/0022-247X(83)90135-X
[12] Brézis, H.: On some degenerate nonlinear parabolic equations. Proc. Symp. Pure Math.18, 28-38 (1970) · Zbl 0231.47034
[13] Brézis, H.: Problèmes unilatéraux. J. Math. Pures Appl.51, 1-168 (1972) · Zbl 0237.35001
[14] Brézis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. Math. Studies 5. Amsterdam: North-Holland 1973 · Zbl 0252.47055
[15] Brézis, H.: Analyse fonctionnelle. Paris: Masson 1983 · Zbl 0511.46001
[16] Browder, F.E.: Pseudo-monotone operators and nonlinear elliptic boundary value problems on unbounded domains. Proc. Natl. Acad. Sci. USA74, 2659-2661 (1977) · Zbl 0358.35034 · doi:10.1073/pnas.74.7.2659
[17] van Duyn, C.J., Hilhorst, D.: On a doubly nonlinear diffusion equation in hydrology. Report 84-26, Department of Mathematics and Informatics, Delft University of Technology, 1984 Nonlinear Anal.11, 305-334 (1987) · Zbl 0654.35049 · doi:10.1016/0362-546X(87)90050-2
[18] Ekeland, I., Temam, R.: Analyse convexe et problèmes variationnels. Paris: Dunod 1974 · Zbl 0281.49001
[19] Esteban, J.R., Vázquez, J.L.: On the equation of turbulent filtration in one-dimensional porous media. Nonlinear Anal.10, 1303-1325 (1986) · Zbl 0613.76102 · doi:10.1016/0362-546X(86)90068-4
[20] Grange, O., Mignot, F.: Sur la résolution d’une équation et d’une inéquation paraboliques non linéaires. J. Funct. Anal.11, 77-92 (1972) · Zbl 0251.35055 · doi:10.1016/0022-1236(72)90080-8
[21] Kalashnikov, A.S.: On a nonlinear equation arising in the theory of nonstationary filtrations. Tr. Semin. Petrovsk.4, 137-146 (1978) (Russian) · Zbl 0415.35044
[22] Kröner, D., Luckhaus, S.: Flow of oil and water in a porous medium. J. Differ. Equations55, 276-288 (1984) · Zbl 0539.35045 · doi:10.1016/0022-0396(84)90084-6
[23] Lions, J.L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Paris: Dunod 1969
[24] Lions, J.L., Magenes, E.: Problèmes aux limites non homogènes et applications. Vol. 1. Paris: Dunod 1968 · Zbl 0165.10801
[25] Mignot, F.: Un théorème d’existence et d’unicité pour une équation parabolique non linéaire. Séminaire Lions-Brézis, 1973-74
[26] Nirenberg, L.: On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa13, 115-162 (1959) · Zbl 0088.07601
[27] Nirenberg, L.: An extended interpolation inequality. Ann. Scuola Norm. Sup. Pisa20, 733-737 (1966) · Zbl 0163.29905
[28] Raviart, P.A.: Sur la résolution de certaines équations paraboliques non linéaires. J. Funct. Anal.5, 299-328 (1970) · Zbl 0199.42401 · doi:10.1016/0022-1236(70)90031-5
[29] Triebel, H.: Interpolation theory, function spaces, differential operators. Amsterdam: North-Holland 1978 · Zbl 0387.46032
[30] Vainberg, M.M.: Variational method and method of monotone operators in the theory of nonlinear equations. New York: Wiley 1973 · Zbl 0279.47022
[31] Barbu, V.: Existence for nonlinear Volterra equations in Hilbert spaces. SIAM J. Math. Anal.10, 552-569 (1979) · Zbl 0462.45021 · doi:10.1137/0510052
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