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2013, 2013(special): 447-456. doi: 10.3934/proc.2013.2013.447

Quasi-subdifferential operators and evolution equations

Masahiro Kubo 1,

1. 

Department of Mathematics, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya 466-8555

Received  August 2012 Revised  December 2012 Published  November 2013

We introduce the concept of a quasi-subdifferential operator and that of a quasi-subdifferential evolution equation. We prove the existence of solutions to related problems and give applications to variational and quasi-variational inequalities.
Keywords: evolution equations., Variational inequalities, subdifferential operators.
Mathematics Subject Classification: Primary: 47J20, 47J35; Secondary 35J87, 35K8.
Citation: Masahiro Kubo. Quasi-subdifferential operators and evolution equations. Conference Publications, 2013, 2013 (special) : 447-456. doi: 10.3934/proc.2013.2013.447
References:
[1]

T. Aiki, Mathematical models including a hysteresis operator, in "Dissipative phase transitions" (eds. P. Colli et al.), Ser. Adv. Math. Appl. Sci. 71 (2006), 1-20.

[2]

H. Attouch, Familles d'operateurs maximaux monotones et mesurabilite, Ann. Mat. Pura Appl. 120 (1979), 35-111.

[3]

H. Attouch, P. Bénilan, A. Damlamian, C. Picard, Equations d'évolution avec condition unilatérale, C. R. Acad. Sci. Paris Ser. A, 279 (1974), 607-609.

[4]

C. Baiocchi and A. Capelo, "Variational and quasivariational inequalities", Wiley-Interscience, Chichester, 1984.

[5]

H. Brézis, Équations et inéquations non linéaires dans les espaces vectoriel en dualité, Ann. Inst. Fourier, 18 (1968), 115-175.

[6]

H. Brézis, Problèmes unilatéraux, J. Math. Pure Appl. IX. Ser., 51 (1972), 1-168.

[7]

H. Brézis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert", North-Holland, Amsterdam-London, 1973.

[8]

F. E. Browder and P. Hess, Nonlinear mappings of monotone type in Banach spaces, J. Funct. Anal., 11 (1972), 251-294.

[9]

P. Colli, N. Kenmochi and M. Kubo, A phase-field model with temperature dependent constraint, J. Math. Anal. Appl., 256 (2001), 668-685.

[10]

J.-L. Joly and U. Mosco, À propos de l'existence et de la régularité des solutions de certaines inéquations quasi-variationnelles, J. Funct. Anal., 34 (1979), 107-137.

[11]

R. Kano, N. Kenmochi and Y. Murase, Elliptic quasi-variational inequalities and applications, Discrete Contin. Dyn. Syst., 2009 Suppl. (2009), 583-591.

[12]

R. Kano, N. Kenmochi and Y. Murase, Parabolic quasi-variational inequalities with nonlocal constraints, Adv. Math. Sci. Appl., 19 (2009), 565-583 .

[13]

R. Kano, Y. Murase and N. Kenmochi, Nonlinear evolution equations generated by subdifferentials with nonlocal constraints Banach Center Publ., 86, Warsaw, 2009, 175-194.

[14]

N. Kenmochi, Some nonlinear parabolic variational inequalities, Israel J. Math., 22 (1975), 304-331.

[15]

N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Educ., Chiba Univ. Part II, 30 (1981), 1-87.

[16]

N. Kenmochi, Monotonicity and compactness methods for nonlinear variational inequalities in "Handbook of Differential Equations" Stationary Partial Differential Equations, Vol. IV (ed. M. Chipot), Elsevier/North Holland, Amsterdam, (2007).

[17]

N. Kenmochi, T. Koyama and G.H. Meyer, Parabolic PDEs with hysteresis and quasivariational inequalities, Nonlinear Anal., 34 (1998), 665-686.

[18]

N. Kenmochi and M. Kubo, Periodic stability of flow in partially saturated porous media, in ''Free Boundary Value Problems", Int. Ser. Numer. Math., 95 (1990), 127-152.

[19]

M. Kubo, Characterization of a class of evolution operators generated by time-dependent subdifferentials, Funkc. Ekvacioj, 32 (1989), 301-321.

[20]

M. Kubo, A filtration model with hysteresis, J. Differ. Equations, 201 (2004), 75-98.

[21]

M. Kubo and N. Yamazaki, Quasilinear parabolic variational inequalities with time-dependent constraints, Adv. Math. Sci. Appl., 15 (2005), 60-68.

[22]

M. Kubo and N. Yamazaki, Elliptic-parabolic variational inequalities with time-dependent constraints, Discrete Contin. Dyn. Syst., 19 (2007), 335-354.

[23]

M. Kubo, K. Shirakawa and N. Yamazaki, Variational inequalities for a system of elliptic-parabolic equations, J. Math. Anal. Appl., 387 (2012), 490-511.

[24]

M. Ôtani, Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators J. Differential Equations, 46 (1982), 268-299.

[25]

M. Ôtani, Nonlinear evolution equations with time-dependent constarints , Adv. Math. Sci. Appl., 3 (): 383. 

[26]

A. Visintin, "Differential models of hysteresis", Springer-Verlag, Berlin, 1994.

[27]

N. Yamazaki, Doubly nonlinear evolution equations associated with elliptic-parabolic free boundary problems, Discrete Contin. Dyn. Syst., 2005 Suppl. (2005), 920-920.

[28]

Y. Yamada, On evolution equations generated by subdifferential operators, J. Fac. Sci., Univ. Tokyo, Sect. IA, 23 (1976), 491-515.

show all references

References:
[1]

T. Aiki, Mathematical models including a hysteresis operator, in "Dissipative phase transitions" (eds. P. Colli et al.), Ser. Adv. Math. Appl. Sci. 71 (2006), 1-20.

[2]

H. Attouch, Familles d'operateurs maximaux monotones et mesurabilite, Ann. Mat. Pura Appl. 120 (1979), 35-111.

[3]

H. Attouch, P. Bénilan, A. Damlamian, C. Picard, Equations d'évolution avec condition unilatérale, C. R. Acad. Sci. Paris Ser. A, 279 (1974), 607-609.

[4]

C. Baiocchi and A. Capelo, "Variational and quasivariational inequalities", Wiley-Interscience, Chichester, 1984.

[5]

H. Brézis, Équations et inéquations non linéaires dans les espaces vectoriel en dualité, Ann. Inst. Fourier, 18 (1968), 115-175.

[6]

H. Brézis, Problèmes unilatéraux, J. Math. Pure Appl. IX. Ser., 51 (1972), 1-168.

[7]

H. Brézis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert", North-Holland, Amsterdam-London, 1973.

[8]

F. E. Browder and P. Hess, Nonlinear mappings of monotone type in Banach spaces, J. Funct. Anal., 11 (1972), 251-294.

[9]

P. Colli, N. Kenmochi and M. Kubo, A phase-field model with temperature dependent constraint, J. Math. Anal. Appl., 256 (2001), 668-685.

[10]

J.-L. Joly and U. Mosco, À propos de l'existence et de la régularité des solutions de certaines inéquations quasi-variationnelles, J. Funct. Anal., 34 (1979), 107-137.

[11]

R. Kano, N. Kenmochi and Y. Murase, Elliptic quasi-variational inequalities and applications, Discrete Contin. Dyn. Syst., 2009 Suppl. (2009), 583-591.

[12]

R. Kano, N. Kenmochi and Y. Murase, Parabolic quasi-variational inequalities with nonlocal constraints, Adv. Math. Sci. Appl., 19 (2009), 565-583 .

[13]

R. Kano, Y. Murase and N. Kenmochi, Nonlinear evolution equations generated by subdifferentials with nonlocal constraints Banach Center Publ., 86, Warsaw, 2009, 175-194.

[14]

N. Kenmochi, Some nonlinear parabolic variational inequalities, Israel J. Math., 22 (1975), 304-331.

[15]

N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Educ., Chiba Univ. Part II, 30 (1981), 1-87.

[16]

N. Kenmochi, Monotonicity and compactness methods for nonlinear variational inequalities in "Handbook of Differential Equations" Stationary Partial Differential Equations, Vol. IV (ed. M. Chipot), Elsevier/North Holland, Amsterdam, (2007).

[17]

N. Kenmochi, T. Koyama and G.H. Meyer, Parabolic PDEs with hysteresis and quasivariational inequalities, Nonlinear Anal., 34 (1998), 665-686.

[18]

N. Kenmochi and M. Kubo, Periodic stability of flow in partially saturated porous media, in ''Free Boundary Value Problems", Int. Ser. Numer. Math., 95 (1990), 127-152.

[19]

M. Kubo, Characterization of a class of evolution operators generated by time-dependent subdifferentials, Funkc. Ekvacioj, 32 (1989), 301-321.

[20]

M. Kubo, A filtration model with hysteresis, J. Differ. Equations, 201 (2004), 75-98.

[21]

M. Kubo and N. Yamazaki, Quasilinear parabolic variational inequalities with time-dependent constraints, Adv. Math. Sci. Appl., 15 (2005), 60-68.

[22]

M. Kubo and N. Yamazaki, Elliptic-parabolic variational inequalities with time-dependent constraints, Discrete Contin. Dyn. Syst., 19 (2007), 335-354.

[23]

M. Kubo, K. Shirakawa and N. Yamazaki, Variational inequalities for a system of elliptic-parabolic equations, J. Math. Anal. Appl., 387 (2012), 490-511.

[24]

M. Ôtani, Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators J. Differential Equations, 46 (1982), 268-299.

[25]

M. Ôtani, Nonlinear evolution equations with time-dependent constarints , Adv. Math. Sci. Appl., 3 (): 383. 

[26]

A. Visintin, "Differential models of hysteresis", Springer-Verlag, Berlin, 1994.

[27]

N. Yamazaki, Doubly nonlinear evolution equations associated with elliptic-parabolic free boundary problems, Discrete Contin. Dyn. Syst., 2005 Suppl. (2005), 920-920.

[28]

Y. Yamada, On evolution equations generated by subdifferential operators, J. Fac. Sci., Univ. Tokyo, Sect. IA, 23 (1976), 491-515.

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