June  2013, 6(3): 771-781. doi: 10.3934/dcdss.2013.6.771

Schrödinger type evolution equations with monotone nonlinearity of non-power type

1. 

Department of Mathematics, Science University of Tokyo, Kagurazaka 1-3, Shinjuku-ku, Tokyo 162-8601, Japan

2. 

Department of Mathematics, Science University of Tokyo, 1-3 Kagurazaka, Sinjuku-ku, Tokyo 162-8601

Received  February 2010 Revised  November 2010 Published  December 2012

Existence of unique strong solutions is established for Schrödinger type evolution equations with monotone nonlinearity. The proof is based on a perturbation theorem for $m$-accretive operators in a complex Hilbert space.
Citation: Yoshiki Maeda, Noboru Okazawa. Schrödinger type evolution equations with monotone nonlinearity of non-power type. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 771-781. doi: 10.3934/dcdss.2013.6.771
References:
[1]

H. Brézis, "Opérateur Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert,", Mathematics Studies, 5 (1973).   Google Scholar

[2]

H. Brézis, "Analyse Fonctionnelle, Théorie et Applications,", Masson, (1983).   Google Scholar

[3]

H. Brézis, M. G. Crandall and A. Pazy, Perturbations of nonlinear maximal monotone sets in Banach spaces,, Comm. Pure Appl. Math., 23 (1970), 123.   Google Scholar

[4]

T. Cazenave, "Semilinear Schrödinger Equations,", Courant Lecture Notes in Math., 10 (2003).   Google Scholar

[5]

N. Ghoussoub, Antisymmetric Hamiltonians$:$ Variational resolutions for Navier-Stokes and other nonlinear evolutions,, Comm. Pure Appl. Math., 60 (2007), 619.  doi: 10.1002/cpa.20176.  Google Scholar

[6]

N. Ghoussoub, "Self-dual Partial Differential Systems and Their Variational Principles,", Springer Monographs in Mathematics, (2009).   Google Scholar

[7]

N. Kita and T. Shimomura, Large time behavior of Schrödinger equations with a dissipative nonlinearity for arbitrarily large initial data,, J. Math. Soc. Japan, 61 (2009), 39.   Google Scholar

[8]

J. L. Lions, "Quelques Méthodes de Résolution des Probl\`emes aux Limites Non Linéaires,", Dunod, (1969).   Google Scholar

[9]

I. Miyadera, "Nonlinear Semigroups,", Translations of Math. Monograph, 109 (1992).   Google Scholar

[10]

N. Okazawa, Sectorialness of second order elliptic operators in divergence form,, Proc. Amer. Math. Soc., 113 (1991), 701.  doi: 10.2307/2048604.  Google Scholar

[11]

N. Okazawa and T. Yokota, Monotonicity method applied to the complex Ginzburg-Landau and related equations,, J. Math. Anal. Appl., 267 (2002), 247.  doi: 10.1006/jmaa.2001.7770.  Google Scholar

[12]

N. Okazawa and T. Yokota, Perturbation theory for $m$-accretive operators and generalized complex Ginzburg-Landau equations,, J. Math. Soc. Japan, 54 (2002), 1.  doi: 10.2969/jmsj/1191593952.  Google Scholar

[13]

N. Okazawa and T. Yokota, Smoothing effect for generalized complex Ginzburg-Landau equations in unbounded domains,, Discrete Contin. Dynam. Systems (2001), Added Volume (2001), 280.   Google Scholar

[14]

N. Okazawa and T. Yokota, Subdifferential operator approach to strong wellposedness of the complex Ginzburg-Landau equation,, Discrete Contin. Dynam. Systems, 28 (2010), 311.  doi: 10.3934/dcds.2010.28.311.  Google Scholar

[15]

H. Pecher and W. von Wahl, Time dependent nonlinear Schrödinger equations,, Manuscripta Math., 27 (1979), 125.  doi: 10.1007/BF01299292.  Google Scholar

[16]

T. Shigeta, A characterization of $m$-accretivity and an application to nonlinear Schrödinger type equations,, Nonlinear Analysis, 10 (1986), 823.  doi: 10.1016/0362-546X(86)90070-2.  Google Scholar

[17]

R. E. Showalter, "Monotone Operators in Banach Space and Nonlinear Partial Differential Equations,", Math. Surv. Mono., 49 (1997).   Google Scholar

[18]

T. Yokota, "Monotonicity and Compactness Methods Applied to the Nonlinear Schrödinger and Related Equations,", Nonlinear analysis and applications: to V. Lakshmikantham on his 80th birthday. 1, 1 (2003), 939.   Google Scholar

show all references

References:
[1]

H. Brézis, "Opérateur Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert,", Mathematics Studies, 5 (1973).   Google Scholar

[2]

H. Brézis, "Analyse Fonctionnelle, Théorie et Applications,", Masson, (1983).   Google Scholar

[3]

H. Brézis, M. G. Crandall and A. Pazy, Perturbations of nonlinear maximal monotone sets in Banach spaces,, Comm. Pure Appl. Math., 23 (1970), 123.   Google Scholar

[4]

T. Cazenave, "Semilinear Schrödinger Equations,", Courant Lecture Notes in Math., 10 (2003).   Google Scholar

[5]

N. Ghoussoub, Antisymmetric Hamiltonians$:$ Variational resolutions for Navier-Stokes and other nonlinear evolutions,, Comm. Pure Appl. Math., 60 (2007), 619.  doi: 10.1002/cpa.20176.  Google Scholar

[6]

N. Ghoussoub, "Self-dual Partial Differential Systems and Their Variational Principles,", Springer Monographs in Mathematics, (2009).   Google Scholar

[7]

N. Kita and T. Shimomura, Large time behavior of Schrödinger equations with a dissipative nonlinearity for arbitrarily large initial data,, J. Math. Soc. Japan, 61 (2009), 39.   Google Scholar

[8]

J. L. Lions, "Quelques Méthodes de Résolution des Probl\`emes aux Limites Non Linéaires,", Dunod, (1969).   Google Scholar

[9]

I. Miyadera, "Nonlinear Semigroups,", Translations of Math. Monograph, 109 (1992).   Google Scholar

[10]

N. Okazawa, Sectorialness of second order elliptic operators in divergence form,, Proc. Amer. Math. Soc., 113 (1991), 701.  doi: 10.2307/2048604.  Google Scholar

[11]

N. Okazawa and T. Yokota, Monotonicity method applied to the complex Ginzburg-Landau and related equations,, J. Math. Anal. Appl., 267 (2002), 247.  doi: 10.1006/jmaa.2001.7770.  Google Scholar

[12]

N. Okazawa and T. Yokota, Perturbation theory for $m$-accretive operators and generalized complex Ginzburg-Landau equations,, J. Math. Soc. Japan, 54 (2002), 1.  doi: 10.2969/jmsj/1191593952.  Google Scholar

[13]

N. Okazawa and T. Yokota, Smoothing effect for generalized complex Ginzburg-Landau equations in unbounded domains,, Discrete Contin. Dynam. Systems (2001), Added Volume (2001), 280.   Google Scholar

[14]

N. Okazawa and T. Yokota, Subdifferential operator approach to strong wellposedness of the complex Ginzburg-Landau equation,, Discrete Contin. Dynam. Systems, 28 (2010), 311.  doi: 10.3934/dcds.2010.28.311.  Google Scholar

[15]

H. Pecher and W. von Wahl, Time dependent nonlinear Schrödinger equations,, Manuscripta Math., 27 (1979), 125.  doi: 10.1007/BF01299292.  Google Scholar

[16]

T. Shigeta, A characterization of $m$-accretivity and an application to nonlinear Schrödinger type equations,, Nonlinear Analysis, 10 (1986), 823.  doi: 10.1016/0362-546X(86)90070-2.  Google Scholar

[17]

R. E. Showalter, "Monotone Operators in Banach Space and Nonlinear Partial Differential Equations,", Math. Surv. Mono., 49 (1997).   Google Scholar

[18]

T. Yokota, "Monotonicity and Compactness Methods Applied to the Nonlinear Schrödinger and Related Equations,", Nonlinear analysis and applications: to V. Lakshmikantham on his 80th birthday. 1, 1 (2003), 939.   Google Scholar

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