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

[1]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317

[2]

Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383

[3]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136

[4]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345

[5]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

[6]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079

[7]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081

[8]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364

[9]

Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020454

[10]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

[11]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432

[12]

Teresa D'Aprile. Bubbling solutions for the Liouville equation around a quantized singularity in symmetric domains. Communications on Pure & Applied Analysis, 2021, 20 (1) : 159-191. doi: 10.3934/cpaa.2020262

[13]

Tomáš Roubíček. Cahn-Hilliard equation with capillarity in actual deforming configurations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 41-55. doi: 10.3934/dcdss.2020303

[14]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296

[15]

Xuemei Chen, Julia Dobrosotskaya. Inpainting via sparse recovery with directional constraints. Mathematical Foundations of Computing, 2020, 3 (4) : 229-247. doi: 10.3934/mfc.2020025

[16]

Andrew D. Lewis. Erratum for "nonholonomic and constrained variational mechanics". Journal of Geometric Mechanics, 2020, 12 (4) : 671-675. doi: 10.3934/jgm.2020033

[17]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[18]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074

[19]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[20]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (34)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]