Quasi-subdifferential operators and evolution equations

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

Quasi-subdifferential operators and evolution equations

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

Received August 2012 Revised December 2012 Published November 2013

Abstract
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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, 71 (2006), 1. Google Scholar
[2]	H. Attouch, Familles d'operateurs maximaux monotones et mesurabilite,, Ann. Mat. Pura Appl. 120* (1979), 120* (1979), 35. Google Scholar
[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. Google Scholar
[4]	C. Baiocchi and A. Capelo, "Variational and quasivariational inequalities",, Wiley-Interscience, (1984). Google Scholar
[5]	H. Brézis, Équations et inéquations non linéaires dans les espaces vectoriel en dualité,, Ann. Inst. Fourier, 18 (1968), 115. Google Scholar
[6]	H. Brézis, Problèmes unilatéraux,, J. Math. Pure Appl. IX. Ser., 51 (1972), 1. Google Scholar
[7]	H. Brézis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert",, North-Holland, (1973). Google Scholar
[8]	F. E. Browder and P. Hess, Nonlinear mappings of monotone type in Banach spaces,, J. Funct. Anal., 11 (1972), 251. Google Scholar
[9]	P. Colli, N. Kenmochi and M. Kubo, A phase-field model with temperature dependent constraint,, J. Math. Anal. Appl., 256 (2001), 668. Google Scholar
[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. Google Scholar
[11]	R. Kano, N. Kenmochi and Y. Murase, Elliptic quasi-variational inequalities and applications,, Discrete Contin. Dyn. Syst., 2009 (2009), 583. Google Scholar
[12]	R. Kano, N. Kenmochi and Y. Murase, Parabolic quasi-variational inequalities with nonlocal constraints,, Adv. Math. Sci. Appl., 19 (2009), 565. Google Scholar
[13]	R. Kano, Y. Murase and N. Kenmochi, Nonlinear evolution equations generated by subdifferentials with nonlocal constraints, Banach Center Publ., 86 (2009), 175. Google Scholar
[14]	N. Kenmochi, Some nonlinear parabolic variational inequalities,, Israel J. Math., 22 (1975), 304. Google Scholar
[15]	N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications,, Bull. Fac. Educ., 30 (1981), 1. Google Scholar
[16]	N. Kenmochi, Monotonicity and compactness methods for nonlinear variational inequalities, in, (2007). Google Scholar
[17]	N. Kenmochi, T. Koyama and G.H. Meyer, Parabolic PDEs with hysteresis and quasivariational inequalities,, Nonlinear Anal., 34 (1998), 665. Google Scholar
[18]	N. Kenmochi and M. Kubo, Periodic stability of flow in partially saturated porous media,, in ''Free Boundary Value Problems, 95 (1990), 127. Google Scholar
[19]	M. Kubo, Characterization of a class of evolution operators generated by time-dependent subdifferentials,, Funkc. Ekvacioj, 32 (1989), 301. Google Scholar
[20]	M. Kubo, A filtration model with hysteresis,, J. Differ. Equations, 201 (2004), 75. Google Scholar
[21]	M. Kubo and N. Yamazaki, Quasilinear parabolic variational inequalities with time-dependent constraints,, Adv. Math. Sci. Appl., 15 (2005), 60. Google Scholar
[22]	M. Kubo and N. Yamazaki, Elliptic-parabolic variational inequalities with time-dependent constraints,, Discrete Contin. Dyn. Syst., 19 (2007), 335. Google Scholar
[23]	M. Kubo, K. Shirakawa and N. Yamazaki, Variational inequalities for a system of elliptic-parabolic equations,, J. Math. Anal. Appl., 387 (2012), 490. Google Scholar
[24]	M. Ôtani, Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators , J. Differential Equations, 46 (1982), 268. Google Scholar
[25]	M. Ôtani, Nonlinear evolution equations with time-dependent constarints , Adv. Math. Sci. Appl., 3 (): 383. Google Scholar
[26]	A. Visintin, "Differential models of hysteresis",, Springer-Verlag, (1994). Google Scholar
[27]	N. Yamazaki, Doubly nonlinear evolution equations associated with elliptic-parabolic free boundary problems,, Discrete Contin. Dyn. Syst., 2005 (2005), 920. Google Scholar
[28]	Y. Yamada, On evolution equations generated by subdifferential operators,, J. Fac. Sci., 23 (1976), 491. Google Scholar

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References:

[1]	T. Aiki, Mathematical models including a hysteresis operator,, in, 71 (2006), 1. Google Scholar
[2]	H. Attouch, Familles d'operateurs maximaux monotones et mesurabilite,, Ann. Mat. Pura Appl. 120* (1979), 120* (1979), 35. Google Scholar
[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. Google Scholar
[4]	C. Baiocchi and A. Capelo, "Variational and quasivariational inequalities",, Wiley-Interscience, (1984). Google Scholar
[5]	H. Brézis, Équations et inéquations non linéaires dans les espaces vectoriel en dualité,, Ann. Inst. Fourier, 18 (1968), 115. Google Scholar
[6]	H. Brézis, Problèmes unilatéraux,, J. Math. Pure Appl. IX. Ser., 51 (1972), 1. Google Scholar
[7]	H. Brézis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert",, North-Holland, (1973). Google Scholar
[8]	F. E. Browder and P. Hess, Nonlinear mappings of monotone type in Banach spaces,, J. Funct. Anal., 11 (1972), 251. Google Scholar
[9]	P. Colli, N. Kenmochi and M. Kubo, A phase-field model with temperature dependent constraint,, J. Math. Anal. Appl., 256 (2001), 668. Google Scholar
[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. Google Scholar
[11]	R. Kano, N. Kenmochi and Y. Murase, Elliptic quasi-variational inequalities and applications,, Discrete Contin. Dyn. Syst., 2009 (2009), 583. Google Scholar
[12]	R. Kano, N. Kenmochi and Y. Murase, Parabolic quasi-variational inequalities with nonlocal constraints,, Adv. Math. Sci. Appl., 19 (2009), 565. Google Scholar
[13]	R. Kano, Y. Murase and N. Kenmochi, Nonlinear evolution equations generated by subdifferentials with nonlocal constraints, Banach Center Publ., 86 (2009), 175. Google Scholar
[14]	N. Kenmochi, Some nonlinear parabolic variational inequalities,, Israel J. Math., 22 (1975), 304. Google Scholar
[15]	N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications,, Bull. Fac. Educ., 30 (1981), 1. Google Scholar
[16]	N. Kenmochi, Monotonicity and compactness methods for nonlinear variational inequalities, in, (2007). Google Scholar
[17]	N. Kenmochi, T. Koyama and G.H. Meyer, Parabolic PDEs with hysteresis and quasivariational inequalities,, Nonlinear Anal., 34 (1998), 665. Google Scholar
[18]	N. Kenmochi and M. Kubo, Periodic stability of flow in partially saturated porous media,, in ''Free Boundary Value Problems, 95 (1990), 127. Google Scholar
[19]	M. Kubo, Characterization of a class of evolution operators generated by time-dependent subdifferentials,, Funkc. Ekvacioj, 32 (1989), 301. Google Scholar
[20]	M. Kubo, A filtration model with hysteresis,, J. Differ. Equations, 201 (2004), 75. Google Scholar
[21]	M. Kubo and N. Yamazaki, Quasilinear parabolic variational inequalities with time-dependent constraints,, Adv. Math. Sci. Appl., 15 (2005), 60. Google Scholar
[22]	M. Kubo and N. Yamazaki, Elliptic-parabolic variational inequalities with time-dependent constraints,, Discrete Contin. Dyn. Syst., 19 (2007), 335. Google Scholar
[23]	M. Kubo, K. Shirakawa and N. Yamazaki, Variational inequalities for a system of elliptic-parabolic equations,, J. Math. Anal. Appl., 387 (2012), 490. Google Scholar
[24]	M. Ôtani, Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators , J. Differential Equations, 46 (1982), 268. Google Scholar
[25]	M. Ôtani, Nonlinear evolution equations with time-dependent constarints , Adv. Math. Sci. Appl., 3 (): 383. Google Scholar
[26]	A. Visintin, "Differential models of hysteresis",, Springer-Verlag, (1994). Google Scholar
[27]	N. Yamazaki, Doubly nonlinear evolution equations associated with elliptic-parabolic free boundary problems,, Discrete Contin. Dyn. Syst., 2005 (2005), 920. Google Scholar
[28]	Y. Yamada, On evolution equations generated by subdifferential operators,, J. Fac. Sci., 23 (1976), 491. Google Scholar

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