# American Institute of Mathematical Sciences

September  2012, 11(5): 2079-2123. doi: 10.3934/cpaa.2012.11.2079

## An abstract existence theorem for parabolic systems

 1 University Bonn, Institute for Applied Mathematics, Endenicher Allee 60, 53115 Bonn, Germany

Received  January 2011 Revised  May 2011 Published  March 2012

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.
Citation: Hans Wilhelm Alt. An abstract existence theorem for parabolic systems. Communications on Pure and Applied Analysis, 2012, 11 (5) : 2079-2123. doi: 10.3934/cpaa.2012.11.2079
##### References:
 [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 [3] H. W. Alt, Partielle Differentialgleichungen III, Vorlesung Winter semester 2003/04, Universität Bonn, unpublished manuscript.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.

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##### References:
 [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.
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