# American Institute of Mathematical Sciences

June  2011, 4(3): 723-744. doi: 10.3934/dcdss.2011.4.723

## Linear evolution equations with strongly measurable families and application to the Dirac equation

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

Received  April 2009 Revised  October 2009 Published  November 2010

A new existence and uniqueness theorem is established for linear evolution equations of hyperbolic type with strongly measurable coefficients in a separable Hilbert space. The result is applied to the Dirac equation with time-dependent potential.
Citation: Noboru Okazawa, Kentarou Yoshii. Linear evolution equations with strongly measurable families and application to the Dirac equation. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 723-744. doi: 10.3934/dcdss.2011.4.723
##### References:
 [1] H. Brézis, "Opérateur maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert," Mathematics Studies 5, North-Holland, Amsterdam, 1973. [2] H. Brézis, "Analyse Fonctionnelle, Théorie et Applications," Masson, Paris, 1983. [3] Calderón, Commutators of singular integral operators, Proc. Nat. Acad. Sci., U.S.A., 53 (1965), 1092-1099. doi: 10.1073/pnas.53.5.1092. [4] G. Da Prato and M. Iannelli, On a method for studying abstract evolution equations in the hyperbolic case, Comm. Partial Differential Equations, 1 (1976), 585-608. doi: 10.1080/03605307608820022. [5] R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology," Vol. 5, Evolution Problem I, Springer-Verlag, Berlin and New York, 1992. [6] H. O. Fattorini, "The Cauchy problem," Encyclopedia Math. Appl., 18, Addison-Wesley, Reading, MA, 1983; Cambridge Univ. Press, New York, 1984. [7] N. Hayashi, P. I. Naumkin and R. B. E. Wibowo, Nonlinear scattering for a system of nonlinear Klein-Gordon equations, J. Math. Phys., 49 (2008), 103501, 24 pp. [8] R. Ikehata and N. Okazawa, Yosida approximation and nonlinear hyperbolic equation, Nonlinear Analysis, TMA, 15 (1990), 479-495. [9] S. Ishii, Linear evolution equations $du/dt+A(t)u=0$: A case where $A(t)$ is strongly uniform-measurable, J. Math. Soc. Japan, 34 (1982), 413-424. doi: 10.2969/jmsj/03430413. [10] T. Kato, "Perturbation Theory for Linear Operators," Grundlehren math. Wissenschaften 132, Springer-Verlag, Berlin and New York, 1966; 2nd ed., 1976. [11] T. Kato, Linear evolution equations of "hyperbolic" type, J. Fac. Sci. Univ. Tokyo, Sec. I., 17 (1970), 241-258. [12] T. Kato, Linear evolution equations of "hyperbolic" type, II, J. Math. Soc. Japan, 25 (1973), 648-666. doi: 10.2969/jmsj/02540648. [13] T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in Lecture Notes in Math., 448, Springer-Verlag, Berlin and New York, 1975, 25-70. [14] T. Kato, Singular perturbation and semigroup theory, in Lecture Notes in Math., 565, Springer-Verlag, Berlin and New York, 1976, 104-112. [15] T. Kato, "Abstract Differential Equations and Nonlinear Mixed Problems," Lezioni Fermiane [Fermi Lectures]. Scuola Normale Superiore, Pisa; Accademia Nazionale dei Lincei, Rome, 1985. [16] T. Kato, Abstract evolution equations, linear and quasilinear, revisited, in Lecture Notes in Math., 1540, Springer-Verlag, Berlin and New York, 1993, 103-125. [17] K. Kobayasi, On a theorem for linear evolution equations of hyperbolic type, J. Math. Soc. Japan, 31 (1979), 647-654. doi: 10.2969/jmsj/03140647. [18] N. Okazawa, Remarks on linear $m$-accretive operators in a Hilbert space, J. Math. Soc. Japan, 27 (1975), 160-165. doi: 10.2969/jmsj/02710160. [19] N. Okazawa, On the perturbation of linear operators in Banach and Hilbert spaces, J. Math. Soc. Japan, 34 (1982), 677-701. doi: 10.2969/jmsj/03440677. [20] N. Okazawa, Abstract quasilinear evolution equations of hyperbolic type, with applications, in "Nonlinear Analysis and Applications" (Warsaw, 1994), GAKUTO Internat. Ser. Math. Sci. Appl., 7, Gakkōtosho, Tokyo, (1996), 303-317. [21] N. Okazawa, Remarks on linear evolution equations of hyperbolic type in Hilbert space, Adv. Math. Sci. Appl., 8 (1998), 399-423. [22] N. Okazawa and A. Unai, Singular perturbation approach to evolution equations of hyperbolic type in Hilbert space, Adv. Math. Sci. Appl., 3 (1993/94), 267-283. [23] N. Okazawa and A. Unai, Linear evolution equations of hyperbolic type in Hilbert space, SUT J. Math., 29 (1993), 51-70. [24] E. M. Ouhabaz, "Analysis of Heat Equations on Domains," London Math. Soc. Monograph, Princeton Univ. Press, Princeton and Oxford, 2005. [25] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Math. Sci., 44, Springer-Verlag, Berlin and New York, 1983. [26] H. Tanabe, "Equations of Evolution," Monographs and Studies in Math., 6, Pitman, London, 1979. [27] H. Tanabe, "Functional Analytic Methods for Partial Differential Equations," Pure and Applied Mathmatics, 204, Marcel Dekker, New York, 1997. [28] N. Tanaka, Nonautonomous abstract Cauchy problems for strongly measurable families, Math. Nachr., 274/275 (2004), 130-153. [29] B. Thaller, "The Dirac Equation," Texts and Monographs in Physics, Springer-Verlag, Berlin and New York, 1992. [30] A. Yagi, On a class of linear evolution equations of "hyperbolic" type in reflexive Banach spaces, Osaka J. Math., 16 (1979), 301-315.

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##### References:
 [1] H. Brézis, "Opérateur maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert," Mathematics Studies 5, North-Holland, Amsterdam, 1973. [2] H. Brézis, "Analyse Fonctionnelle, Théorie et Applications," Masson, Paris, 1983. [3] Calderón, Commutators of singular integral operators, Proc. Nat. Acad. Sci., U.S.A., 53 (1965), 1092-1099. doi: 10.1073/pnas.53.5.1092. [4] G. Da Prato and M. Iannelli, On a method for studying abstract evolution equations in the hyperbolic case, Comm. Partial Differential Equations, 1 (1976), 585-608. doi: 10.1080/03605307608820022. [5] R. Dautray and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology," Vol. 5, Evolution Problem I, Springer-Verlag, Berlin and New York, 1992. [6] H. O. Fattorini, "The Cauchy problem," Encyclopedia Math. Appl., 18, Addison-Wesley, Reading, MA, 1983; Cambridge Univ. Press, New York, 1984. [7] N. Hayashi, P. I. Naumkin and R. B. E. Wibowo, Nonlinear scattering for a system of nonlinear Klein-Gordon equations, J. Math. Phys., 49 (2008), 103501, 24 pp. [8] R. Ikehata and N. Okazawa, Yosida approximation and nonlinear hyperbolic equation, Nonlinear Analysis, TMA, 15 (1990), 479-495. [9] S. Ishii, Linear evolution equations $du/dt+A(t)u=0$: A case where $A(t)$ is strongly uniform-measurable, J. Math. Soc. Japan, 34 (1982), 413-424. doi: 10.2969/jmsj/03430413. [10] T. Kato, "Perturbation Theory for Linear Operators," Grundlehren math. Wissenschaften 132, Springer-Verlag, Berlin and New York, 1966; 2nd ed., 1976. [11] T. Kato, Linear evolution equations of "hyperbolic" type, J. Fac. Sci. Univ. Tokyo, Sec. I., 17 (1970), 241-258. [12] T. Kato, Linear evolution equations of "hyperbolic" type, II, J. Math. Soc. Japan, 25 (1973), 648-666. doi: 10.2969/jmsj/02540648. [13] T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in Lecture Notes in Math., 448, Springer-Verlag, Berlin and New York, 1975, 25-70. [14] T. Kato, Singular perturbation and semigroup theory, in Lecture Notes in Math., 565, Springer-Verlag, Berlin and New York, 1976, 104-112. [15] T. Kato, "Abstract Differential Equations and Nonlinear Mixed Problems," Lezioni Fermiane [Fermi Lectures]. Scuola Normale Superiore, Pisa; Accademia Nazionale dei Lincei, Rome, 1985. [16] T. Kato, Abstract evolution equations, linear and quasilinear, revisited, in Lecture Notes in Math., 1540, Springer-Verlag, Berlin and New York, 1993, 103-125. [17] K. Kobayasi, On a theorem for linear evolution equations of hyperbolic type, J. Math. Soc. Japan, 31 (1979), 647-654. doi: 10.2969/jmsj/03140647. [18] N. Okazawa, Remarks on linear $m$-accretive operators in a Hilbert space, J. Math. Soc. Japan, 27 (1975), 160-165. doi: 10.2969/jmsj/02710160. [19] N. Okazawa, On the perturbation of linear operators in Banach and Hilbert spaces, J. Math. Soc. Japan, 34 (1982), 677-701. doi: 10.2969/jmsj/03440677. [20] N. Okazawa, Abstract quasilinear evolution equations of hyperbolic type, with applications, in "Nonlinear Analysis and Applications" (Warsaw, 1994), GAKUTO Internat. Ser. Math. Sci. Appl., 7, Gakkōtosho, Tokyo, (1996), 303-317. [21] N. Okazawa, Remarks on linear evolution equations of hyperbolic type in Hilbert space, Adv. Math. Sci. Appl., 8 (1998), 399-423. [22] N. Okazawa and A. Unai, Singular perturbation approach to evolution equations of hyperbolic type in Hilbert space, Adv. Math. Sci. Appl., 3 (1993/94), 267-283. [23] N. Okazawa and A. Unai, Linear evolution equations of hyperbolic type in Hilbert space, SUT J. Math., 29 (1993), 51-70. [24] E. M. Ouhabaz, "Analysis of Heat Equations on Domains," London Math. Soc. Monograph, Princeton Univ. Press, Princeton and Oxford, 2005. [25] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Math. Sci., 44, Springer-Verlag, Berlin and New York, 1983. [26] H. Tanabe, "Equations of Evolution," Monographs and Studies in Math., 6, Pitman, London, 1979. [27] H. Tanabe, "Functional Analytic Methods for Partial Differential Equations," Pure and Applied Mathmatics, 204, Marcel Dekker, New York, 1997. [28] N. Tanaka, Nonautonomous abstract Cauchy problems for strongly measurable families, Math. Nachr., 274/275 (2004), 130-153. [29] B. Thaller, "The Dirac Equation," Texts and Monographs in Physics, Springer-Verlag, Berlin and New York, 1992. [30] A. Yagi, On a class of linear evolution equations of "hyperbolic" type in reflexive Banach spaces, Osaka J. Math., 16 (1979), 301-315.
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