American Institute of Mathematical Sciences

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

An abstract existence theorem for parabolic systems

Hans Wilhelm Alt 1,

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.
Keywords: existence theorem, Parabolic equations, abstract theory..
Mathematics Subject Classification: 35D30, 35K59, 35K65, 35K90, 35R37, 47F05, 47J30, 47J35, 47H0.
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

show all references

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

[1]

Wolf-Jürgen Beyn, Sergey Piskarev. Shadowing for discrete approximations of abstract parabolic equations. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 19-42. doi: 10.3934/dcdsb.2008.10.19
[2]

Hernán R. Henríquez, Claudio Cuevas, Juan C. Pozo, Herme Soto. Existence of solutions for a class of abstract neutral differential equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2455-2482. doi: 10.3934/dcds.2017106
[3]

Luisa Arlotti, Bertrand Lods, Mustapha Mokhtar-Kharroubi. Non-autonomous Honesty theory in abstract state spaces with applications to linear kinetic equations. Communications on Pure & Applied Analysis, 2014, 13 (2) : 729-771. doi: 10.3934/cpaa.2014.13.729
[4]

Takeshi Fukao, Nobuyuki Kenmochi. Abstract theory of variational inequalities and Lagrange multipliers. Conference Publications, 2013, 2013 (special) : 237-246. doi: 10.3934/proc.2013.2013.237
[5]

Mehdi Badra. Abstract settings for stabilization of nonlinear parabolic system with a Riccati-based strategy. Application to Navier-Stokes and Boussinesq equations with Neumann or Dirichlet control. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1169-1208. doi: 10.3934/dcds.2012.32.1169
[6]

Begoña Barrios, Leandro Del Pezzo, Jorge García-Melián, Alexander Quaas. A Liouville theorem for indefinite fractional diffusion equations and its application to existence of solutions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5731-5746. doi: 10.3934/dcds.2017248
[7]

Maria Michaela Porzio. Existence of solutions for some "noncoercive" parabolic equations. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 553-568. doi: 10.3934/dcds.1999.5.553
[8]

Daniela Giachetti, Maria Michaela Porzio. Global existence for nonlinear parabolic equations with a damping term. Communications on Pure & Applied Analysis, 2009, 8 (3) : 923-953. doi: 10.3934/cpaa.2009.8.923
[9]

Tôn Việt Tạ. Existence results for linear evolution equations of parabolic type. Communications on Pure & Applied Analysis, 2018, 17 (3) : 751-785. doi: 10.3934/cpaa.2018039
[10]

Manuel Torrilhon. H-Theorem for nonlinear regularized 13-moment equations in kinetic gas theory. Kinetic & Related Models, 2012, 5 (1) : 185-201. doi: 10.3934/krm.2012.5.185
[11]

Noboru Okazawa, Toshiyuki Suzuki, Tomomi Yokota. Energy methods for abstract nonlinear Schrödinger equations. Evolution Equations & Control Theory, 2012, 1 (2) : 337-354. doi: 10.3934/eect.2012.1.337
[12]

Valeria Danese, Pelin G. Geredeli, Vittorino Pata. Exponential attractors for abstract equations with memory and applications to viscoelasticity. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2881-2904. doi: 10.3934/dcds.2015.35.2881
[13]

Hiroshi Watanabe. Existence and uniqueness of entropy solutions to strongly degenerate parabolic equations with discontinuous coefficients. Conference Publications, 2013, 2013 (special) : 781-790. doi: 10.3934/proc.2013.2013.781
[14]

B. Abdellaoui, E. Colorado, I. Peral. Existence and nonexistence results for a class of parabolic equations with mixed boundary conditions. Communications on Pure & Applied Analysis, 2006, 5 (1) : 29-54. doi: 10.3934/cpaa.2006.5.29
[15]

Dominique Blanchard, Olivier Guibé, Hicham Redwane. Existence and uniqueness of a solution for a class of parabolic equations with two unbounded nonlinearities. Communications on Pure & Applied Analysis, 2016, 15 (1) : 197-217. doi: 10.3934/cpaa.2016.15.197
[16]

Peiying Chen. Existence and uniqueness of weak solutions for a class of nonlinear parabolic equations. Electronic Research Announcements, 2017, 24: 38-52. doi: 10.3934/era.2017.24.005
[17]

Olaf Hansen. A global existence theorem for two coupled semilinear diffusion equations from climate modeling. Discrete & Continuous Dynamical Systems - A, 1997, 3 (4) : 541-564. doi: 10.3934/dcds.1997.3.541
[18]

Zhihua Huang, Xiaochun Liu. Existence theorem for a class of semilinear totally characteristic elliptic equations involving supercritical cone sobolev exponents. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3201-3216. doi: 10.3934/cpaa.2019144
[19]

Yukie Goto, Danielle Hilhorst, Ehud Meron, Roger Temam. Existence theorem for a model of dryland vegetation. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 197-224. doi: 10.3934/dcdsb.2011.16.197
[20]

Wenying Feng, Guang Zhang, Yikang Chai. Existence of positive solutions for second order differential equations arising from chemical reactor theory. Conference Publications, 2007, 2007 (Special) : 373-381. doi: 10.3934/proc.2007.2007.373

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences