Article Contents
Article Contents

# An abstract existence theorem for parabolic systems

• In this paper we prove an abstract existence theorem which can be applied to solve parabolic problems in a wide range of applications. It also applies to parabolic variational inequalities. The abstract theorem is based on a Gelfand triple $(V,H,V^*)$, where the standard realization for parabolic systems of second order is $(W^{1, 2}(\Omega),L^2(\Omega), W^{1,2}(\Omega)^*)$. But also realizations to other problems are possible, for example, to fourth order systems.
In all applications to boundary value problems the set $M\subset V$ is an affine subspace, whereas for variational inequalities the constraint $M$ is a closed convex set.
The proof is purely abstract and new.
The corresponding compactness theorem is based on [5].
The present paper is suitable for lectures, since it relays on the corresponding abstract elliptic theory.
Mathematics Subject Classification: 35D30, 35K59, 35K65, 35K90, 35R37, 47F05, 47J30, 47J35, 47H05.

 Citation:

•  [1] H. Amann, "Linear and Quasilinear Parabolic Problems, Volume I: Abstract Linear Theory," Monograph in Mathematics, Birkhäuser Basel, 1995. [2] H. W. Alt, "Elliptische Probleme mit freiem Rand," Lecture Notes 21 SFB 256, Bonn, 1991. [3] H. W. Alt, Partielle Differentialgleichungen III, Vorlesung Winter semester 2003/04, Universität Bonn, unpublished manuscript. [4] H. W. Alt and E. DiBenedetto, Nonsteady flow of water and oil through inhomogeneous porous media, Ann. Scuola Norm. Sup. Pisa, Cl. Sci., 12 (1985), 335-392. [5] H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 183 (1983), 311-341. [6] H. W. Alt, S. Luckhaus and A. Visintin, On nonstationary flow through porous media, Ann. Mat. Pura Appl., 136 (1984), 303-316. [7] A.-K. Becher, "Ein abstrakter Existenzsatz für elliptisch-parabolische Systeme," Diplomarbeit 2005, Universität Bonn. [8] M. S. Berger, "Nonlinearity and Functional Analysis," Lectures on Nonlinear Problems in Mathematical Analysis, Academic Press, 1977. [9] F. Bernis, Existence results for doubly nonlinear higher order parabolic equations on unbounded domains, Math. Ann., 279 (1988), 373-394. [10] E. DiBenedetto, "Degenerate Parabolic Equations," Universitext, Springer, 1993. [11] G. Duvaut and J. L. Lions, "Inequalities in Mechanics and Physics," Grundlehren der mathematischen Wissenschaften 219, Springer-Verlag, 1976. [12] A. Friedman, "Partial Differential Equations," Holt, Rinehart and Winston New York, 1969. [13] U. Fermum, "Nichtlineare elliptisch-parabolische Gleichungen mit zeitabhängigen Hindernissen," Diplomarbeit 2005, Universität Bonn. [14] W. Jäger and J. Kačur, Solution of doubly nonlinear and degenerate parabolic problems by relaxation schemes, Math. Modelling Numer. Anal., 29 (1995), 605-627. [15] N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and application, Bull. Fac. Education Chiba Univ., 30 (1981), 1-87. [16] N. Kenmochi and I. Pawlow, A class of nonlinear elliptic-parabolic equations with time dependent constraints, Nonlinear Analysis, 10 (1986), 1181-1202. [17] D. Kinderlehrer and G. Stampacchia, "An Introduction to Variational Inequalities and their Application," Academic Press, 1980. [18] D. Kröner and S. Luckhaus, Flow of oil and water in a porous medium, J. Differential Equations, 55 (1984), 276-288. [19] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Translations of Mathematical Monographs, Vol. 23, 1968. [20] I. Müller, "Thermodynamics," Interaction of mechanics and mathematics series, Pitman Boston London Melbourne, 1985. [21] P. A. Raviart, Sur la résolution de certaines equations paraboliques non linéaires, J. Functional Analysis, 5 (1970), 299-328. [22] M. Růžička, "Nichtlineare Funktionalanalysis. Eine Einführung," See also the version in http://aam.mathematik.uni-freiburg.de/IAM/homepages/rose/springer.html, Springer, 2004. [23] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987). [24] M. Schmidt, "Ein Existenzsatz für parabolische Systeme zur Beschreibung von chemischen Reaktionen," Diplomarbeit 2006, Universität Bonn. [25] A. Visintin, "Models of Phase Transition," Birkhäuser, 1996. [26] G. Duvaut and J. L. Lions, "Inequalities in Mechanics and Physics," Grundlehren der mathematischen Wissenschaften 219, Springer-Verlag, 1976. [27] E. DiBenedetto, "Partial Differential Equations," 2nd edition, Birkhäuser Boston, 2010.