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Global solvability of second order evolution equations in banach scales

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Calculus of Variations and Partial Differential Equations

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1340))

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References

  1. A. Arosio, Asymptotic behaviour as t→+∞ of the solutions of linear hyperbolic equations with coefficients discontinuous in time (on a bounded domain), J. Differential Equations 39 (1981), no. 2, 291–309.

    Article  MathSciNet  MATH  Google Scholar 

  2. A. Arosio, Linear second order differential equations in Hilbert space. The Cauchy problem and asymptotic behaviour for large time, Arch. Rational Mech. Anal. 86 (1984), no. 2, 147–180; Equations differentielles operationnelles lineaires du deuxieme ordre: problème de Cauchy et comportement asymptotique lorsque t→+∞, C.R. Acad. Sci. Paris, Ser. I, 295 (1982), 83–86.

    Article  MathSciNet  MATH  Google Scholar 

  3. A. Arosio, Abstract linear hyperbolic equations with variable domain, Ann. Mat. Pura Appl. 135 (1983), 173–218.

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Arosio and S. Spagnolo, Global solutions of the Cauchy problem for a non-linear hyperbolic equation, Nonlinear Partial Differential Equations and their applications, College de France, Seminar 6, 1–26; H. Brezis and J.L. Lions ed., Research Notes Math. 109, Pitman, Boston 1984.

    Google Scholar 

  5. A. Arosio and S. Spagnolo, Global existence for abstract evolution equations of weakly hyperbolic type, J. Math. Pures Appl. 65 (1986), 1–43.

    MathSciNet  MATH  Google Scholar 

  6. S. Bernstein, Sur une classe d’equations fonctionnelles aux derivees partielles, Izv. Akad. Nauk SSSR, Ser. Mat. 4 (1940), 17–26.

    MathSciNet  MATH  Google Scholar 

  7. J.-M. Bony and P. Schapira, Existence et prolongement des solutions holomorphes des equations aux derivees partielles, Invent. Math. 17 (1972), 95–105.

    Article  MathSciNet  MATH  Google Scholar 

  8. L. Cardosi, Evolution equations in scales of abstract Gevrey spaces, Boll. Un. Mat. Ital., Ser. VI, 4-C, no. 1 (1985), 379–406.

    MathSciNet  MATH  Google Scholar 

  9. G.F. Carrier, On the non-linear vibration problem of the elastic string, Quart. Appl. Math. 3 (1945), 157–165; A note on the vibrating sring, ibidem 7 (1949), 97–101.

    MathSciNet  MATH  Google Scholar 

  10. F. Colombini, E. De Giorgi and S. Spagnolo, Sur les equations hyperboliques avec des coefficients qui ne dependent que du temps, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 6 (1979), 511–559.

    MathSciNet  MATH  Google Scholar 

  11. F. Colombini and S. Spagnolo, Second order hyperbolic equations with coefficients real analytic in space variables and discontinuous in time, J. Analyse Math. 38 (1980), 1–33.

    Article  MathSciNet  MATH  Google Scholar 

  12. F. Colombini and S. Spagnolo, An example of a weakly hyperbolic Cauchy problem not well posed in C∞, Acta Math. 148 (1982), 243–253.

    Article  MathSciNet  MATH  Google Scholar 

  13. V.I. Derguzov and V.A. Jakubovič, Existence of solutions of linear Hamilton equations with unbounded operator coefficients, Dokl. Akad. Nauk SSSR 151 (1963), no. 6, 1264–1267 (transl.: Soviet Math. Dokl. 4 (1963), 1169–1172.

    MathSciNet  MATH  Google Scholar 

  14. L. De Simon and G. Torelli, Linear second order differential equations with discontinuous coefficients in Hilbert spaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 1 (1974), 131–154.

    MathSciNet  MATH  Google Scholar 

  15. E. Jannelli, Weakly hyperbolic equations of second order with coefficients real analytic in space variables, Comm. Partial Differential Equations 7 (1982), 537–558.

    Article  MathSciNet  MATH  Google Scholar 

  16. T. Kato, Linear and quasi-linear equations of evolution of hyperbolic type, Hyperbolicity, C.I.M.E. Cortona, 1976, Liguori, Napoli (1977), 127–192.

    Google Scholar 

  17. J.L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Dunod, Paris 1968.

    MATH  Google Scholar 

  18. S. Mizohata, Analyticity of solutions of hyperbolic systems with analytic coefficients, Comm. Pure Appl. Math. 14 (1961), 547–559.

    Article  MathSciNet  MATH  Google Scholar 

  19. R. Narasimha, Nonlinear vibration of an elastic string, J. Sound Vibration 8 (1968), 134–146.

    Article  MATH  Google Scholar 

  20. L.V. Ovsjannikov, A singular operator in a scale of Banach spaces, Dokl. Akad. Nauk SSSR 163 no. 4 (1965), 819–822 (transl.: Soviet Math. Dokl. 6 (1965), 1025–1028).

    MathSciNet  Google Scholar 

  21. S.I. Pohožaev, On a class of quasilinear hyperbolic equations, Mat. Sbornik 96 (138), no. 1 (1975), 152–166 (transl.: Math. USSR Sbornik 25, no. 1 (1975), 145–158).

    MathSciNet  Google Scholar 

  22. T. Yamanaka, Note on Kowalevskaja’s system of partial differential equations, Comment.

    Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica - Università, Via dell’Università, 43100, Parma, Italy

    Alberto Arosio

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  1. Alberto Arosio
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Stefan Hildebrandt David Kinderlehrer Mario Miranda

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Dedicated to Hans Lewy

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© 1988 Springer-Verlag

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Arosio, A. (1988). Global solvability of second order evolution equations in banach scales. In: Hildebrandt, S., Kinderlehrer, D., Miranda, M. (eds) Calculus of Variations and Partial Differential Equations. Lecture Notes in Mathematics, vol 1340. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0082880

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  • DOI: https://doi.org/10.1007/BFb0082880

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  • Print ISBN: 978-3-540-50119-0

  • Online ISBN: 978-3-540-45932-3

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