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

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