February  2014, 7(1): 75-93. doi: 10.3934/dcdss.2014.7.75

Solvability of nonlinear evolution equations generated by subdifferentials and perturbations

1. 

Faculty of Education, Kochi University, 2-5-1 Akebono-cho, Kochi 780-8520, Japan

2. 

Department of Mathematics, Faculty of Science and Technology, Meijo University, 1-501 Shiogamaguchi, Tenpaku-ku, Nagoya, 468-8502

Received  March 2012 Revised  October 2012 Published  July 2013

The main objective of this paper is to discuss solvability of the Cauchy problem of an evolution equation with subdifferentials of convex functions which is generated by unknown functions and perturbations of the form:
     $  u'(t) + ∂ \varphi^t(u;u(t)) + G(u(t)) \ni f(t) $   0 < t < T,      in     H.
where H is a Hilbert space, $u'=\frac{du}{dt}$, and $∂ \varphi^t(u;\cdot )$ is a subdifferential operator of convex function $\varphi^t(u;\cdot )$. The evolution equation corresponds to parabolic quasi-variational inequalities.
Citation: Risei Kano, Yusuke Murase. Solvability of nonlinear evolution equations generated by subdifferentials and perturbations. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 75-93. doi: 10.3934/dcdss.2014.7.75
References:
[1]

A. Attouch, "Variational Convergence for Functions and Operators,", Applicable Mathematics Series, (1984). Google Scholar

[2]

A. Azevedo and L. Santos, Convergence of convex sets with gradient constraint,, J. Convex Anal., 11 (2004), 285. Google Scholar

[3]

M. Biroli, Sur les inéquations paraboliques avec convexe dépendant du temps: Solution forte et solution faible,, Riv. Mat., 3 (1974), 33. Google Scholar

[4]

H. Brézis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert,", (French) North-Holland Mathematics Studies, (1973). Google Scholar

[5]

H. Brézis, Problémes unilatéraux,, (French) J. Math. Pures Appl., 51 (1972), 1. Google Scholar

[6]

K. H. Hoffmann, M. Kubo and N. Yamazaki, Optimal control problems for elliptic-parabolic variational inequalities with time-dependent constraints,, Numer. Funct. Anal. Optim., 27 (2006), 329. doi: 10.1080/01630560600686116. Google Scholar

[7]

A. Kadoya, N. Kenmochi and Y. Murase, A class of nonlinear parabolic systems with environmental constraints,, Adv. Math. Sci. Appl., 20 (2010), 281. Google Scholar

[8]

R. Kano, Applications of abstract parabolic quasi-variational inequalities to obstacle problems,, in, 86 (2009), 163. doi: 10.4064/bc86-0-10. Google Scholar

[9]

R. Kano and A. Ito, The existence of time global solutions for tumor invasion models with constraints,, Discrete Contin. Dyn. Syst., 2011 (): 774. Google Scholar

[10]

R. Kano, A. Ito, K. Yamamoto and H. Nakayama, Quasi-variational inequality approach to tumor invasion models with constraints,, GAKUTO Internat. Ser. Math. Sci. Appl., 32 (2010), 365. Google Scholar

[11]

R. Kano, N. Kenmochi and Y. Murase, Existence theorems for elliptic quasi-variational inequalities in Banach spaces,, in, (2008), 149. doi: 10.1142/9789812709257_0010. Google Scholar

[12]

R. Kano, N. Kenmochi and Y. Murase, Nonlinear evolution equations generated by subdifferentials with nonlocal constraints,, in, 86 (2009), 175. doi: 10.4064/bc86-0-11. Google Scholar

[13]

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

[14]

R. Kano, N. Kenmochi and Y. Murase, Elliptic quasi-variational inequalities and applications,, Discrete Contin. Dyn. Syst., 2009 (): 583. doi: 10.4064/bc86-0-15. Google Scholar

[15]

N. Kenmochi, Some nonlinear parabolic variational inequalities,, Israel J. Math., 22 (1975), 304. doi: 10.1007/BF02761596. Google Scholar

[16]

N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications,, Bull. Fac. Education, 30 (1981), 1. Google Scholar

[17]

N. Kenmochi, "Monotonicity and compactness methods for nonlinear variational inequalities,", in, (2007), 203. doi: 10.1016/S1874-5733(07)80007-6. Google Scholar

[18]

M. Kunze and J. F. Rodrigues, An elliptic quasi-variational inequality with gradient constraints and some of its applications,, Math. Methods Appl. Sci., 23 (2000), 897. doi: 10.1002/1099-1476(20000710)23:10<897::AID-MMA141>3.0.CO;2-H. Google Scholar

[19]

F. Mignot and J.-P. Puel, Inéquations d'évolution paraboliques avec convexes d'épe- ndant du temps. Applications aux inéquations quasi variationnelles d'évolution,, Arch. Rat. Mech. Anal., 64 (1977), 59. doi: 10.1007/BF00280179. Google Scholar

[20]

U. Mosco, Convergence of convex sets and of solutions of variational inequalities,, Advances in Math., 3 (1969), 510. doi: 10.1016/0001-8708(69)90009-7. Google Scholar

[21]

Y. Murase, Abstract quasi-variational inequalities of elliptic type and applications,, in, 86 (2009), 235. doi: 10.4064/bc86-0-15. Google Scholar

[22]

J. F. Rodrigues, On a quasi-variational inequality arising in semiconductor theory,, Revista Mat., 5 (1992), 137. Google Scholar

[23]

J. F. Rodrigues and L. Santos, A parabolic quasi-variational inequality arising in a superconductivity model,, Ann. Scuola Norm. Sup. Pisa, 29 (2000), 153. Google Scholar

[24]

U. Stefanelli, Nonlocal quasivariational evolution problems,, J. Differential Equations, 229 (2006), 204. doi: 10.1016/j.jde.2006.05.004. Google Scholar

[25]

Y. Yamada, On nonlinear evolution equations generated by the subdifferentials,, J. Fac. Sci., 23 (1976), 491. Google Scholar

show all references

References:
[1]

A. Attouch, "Variational Convergence for Functions and Operators,", Applicable Mathematics Series, (1984). Google Scholar

[2]

A. Azevedo and L. Santos, Convergence of convex sets with gradient constraint,, J. Convex Anal., 11 (2004), 285. Google Scholar

[3]

M. Biroli, Sur les inéquations paraboliques avec convexe dépendant du temps: Solution forte et solution faible,, Riv. Mat., 3 (1974), 33. Google Scholar

[4]

H. Brézis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert,", (French) North-Holland Mathematics Studies, (1973). Google Scholar

[5]

H. Brézis, Problémes unilatéraux,, (French) J. Math. Pures Appl., 51 (1972), 1. Google Scholar

[6]

K. H. Hoffmann, M. Kubo and N. Yamazaki, Optimal control problems for elliptic-parabolic variational inequalities with time-dependent constraints,, Numer. Funct. Anal. Optim., 27 (2006), 329. doi: 10.1080/01630560600686116. Google Scholar

[7]

A. Kadoya, N. Kenmochi and Y. Murase, A class of nonlinear parabolic systems with environmental constraints,, Adv. Math. Sci. Appl., 20 (2010), 281. Google Scholar

[8]

R. Kano, Applications of abstract parabolic quasi-variational inequalities to obstacle problems,, in, 86 (2009), 163. doi: 10.4064/bc86-0-10. Google Scholar

[9]

R. Kano and A. Ito, The existence of time global solutions for tumor invasion models with constraints,, Discrete Contin. Dyn. Syst., 2011 (): 774. Google Scholar

[10]

R. Kano, A. Ito, K. Yamamoto and H. Nakayama, Quasi-variational inequality approach to tumor invasion models with constraints,, GAKUTO Internat. Ser. Math. Sci. Appl., 32 (2010), 365. Google Scholar

[11]

R. Kano, N. Kenmochi and Y. Murase, Existence theorems for elliptic quasi-variational inequalities in Banach spaces,, in, (2008), 149. doi: 10.1142/9789812709257_0010. Google Scholar

[12]

R. Kano, N. Kenmochi and Y. Murase, Nonlinear evolution equations generated by subdifferentials with nonlocal constraints,, in, 86 (2009), 175. doi: 10.4064/bc86-0-11. Google Scholar

[13]

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

[14]

R. Kano, N. Kenmochi and Y. Murase, Elliptic quasi-variational inequalities and applications,, Discrete Contin. Dyn. Syst., 2009 (): 583. doi: 10.4064/bc86-0-15. Google Scholar

[15]

N. Kenmochi, Some nonlinear parabolic variational inequalities,, Israel J. Math., 22 (1975), 304. doi: 10.1007/BF02761596. Google Scholar

[16]

N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications,, Bull. Fac. Education, 30 (1981), 1. Google Scholar

[17]

N. Kenmochi, "Monotonicity and compactness methods for nonlinear variational inequalities,", in, (2007), 203. doi: 10.1016/S1874-5733(07)80007-6. Google Scholar

[18]

M. Kunze and J. F. Rodrigues, An elliptic quasi-variational inequality with gradient constraints and some of its applications,, Math. Methods Appl. Sci., 23 (2000), 897. doi: 10.1002/1099-1476(20000710)23:10<897::AID-MMA141>3.0.CO;2-H. Google Scholar

[19]

F. Mignot and J.-P. Puel, Inéquations d'évolution paraboliques avec convexes d'épe- ndant du temps. Applications aux inéquations quasi variationnelles d'évolution,, Arch. Rat. Mech. Anal., 64 (1977), 59. doi: 10.1007/BF00280179. Google Scholar

[20]

U. Mosco, Convergence of convex sets and of solutions of variational inequalities,, Advances in Math., 3 (1969), 510. doi: 10.1016/0001-8708(69)90009-7. Google Scholar

[21]

Y. Murase, Abstract quasi-variational inequalities of elliptic type and applications,, in, 86 (2009), 235. doi: 10.4064/bc86-0-15. Google Scholar

[22]

J. F. Rodrigues, On a quasi-variational inequality arising in semiconductor theory,, Revista Mat., 5 (1992), 137. Google Scholar

[23]

J. F. Rodrigues and L. Santos, A parabolic quasi-variational inequality arising in a superconductivity model,, Ann. Scuola Norm. Sup. Pisa, 29 (2000), 153. Google Scholar

[24]

U. Stefanelli, Nonlocal quasivariational evolution problems,, J. Differential Equations, 229 (2006), 204. doi: 10.1016/j.jde.2006.05.004. Google Scholar

[25]

Y. Yamada, On nonlinear evolution equations generated by the subdifferentials,, J. Fac. Sci., 23 (1976), 491. Google Scholar

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