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Linear second order differential equations in hilbert spaces

The Cauchy Problem and asymptotic behaviour for large time

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References

  1. L. Amerio & G. Prouse, “Almost periodic functions and functional equations” Van Nostrand Reinhold C., New York, 1971.

    Google Scholar 

  2. B. G. Ararkcjan, Departure to almost periodic behaviour of solutions of boundary value problems for a hyperbolic equation, Trudy Mat. Inst. Steklov 103 (1968), 5–14 (transl. in: Proc. Steklov Inst. Mat. 103 (1968), 1–12).

    Google Scholar 

  3. A. Arosio, Asymptotic behaviour as t → + ∞ of the solutions of linear hyperbolic equations with coefficients discontinuous in time (on a bounded domain), J. Diff. Eq. 39 (1981), 291–309.

    Google Scholar 

  4. G. Ascoli, Sulla forma asintotica degli integrali dell'equazione ÿ + A(x) y = 0 in un caso notevole di stabilità, Univ. Nac. Tucumán, Revista A 2 (1941), 131–140 (MR 4 #43).

    Google Scholar 

  5. C. Baiocchi, Soluzioni ordinarie e generalizzate del problema di Cauchy per equazioni differenziali astratte lineari del secondo ordine in spazi di Hilbert, Ricerche Mat. XVI(1967), 27–95.

    Google Scholar 

  6. H. Brezis “Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert” North-Holland, London, 1973.

    Google Scholar 

  7. E. A. Coddington & N. Levinson, “Theory of ordinary differential equations” McGraw-Hill B.C., New York, 1955.

    Google Scholar 

  8. F. Colombini, E. de Giorgi & S. Spagnolo, Sur les équations hyperboliques avec de coefficients qui ne dépendent que du temps, Ann. Sc. Norm. Sup. Pisa, serie IV, 6 (1979), 511–559.

    Google Scholar 

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

    Google Scholar 

  10. Ju. L. Daleckii, “Investigations connected with operator evolution equation (functional integrals. Asymptotic integration)” (Russian), Author's review of Doctoral Dissertation, Moscow State Univ., Moscow, 1962.

    Google Scholar 

  11. G. da Prato, Weak solution for linear abstract differential equations in Banach spaces, Advances in Math. 5 (1970), 181–245.

    Google Scholar 

  12. G. da Prato & M. Iannelli, On a method for studying abstract evolution equations in the hyperbolic case, Comm. Part. Diff. Eq. 1 (1976), 585–608.

    Google Scholar 

  13. V. I. Derguzov & V. A. Jakubovič, Existence of solutions of linear Hamilton equations with unbounded operator coefficients, Dokl. Akad. Nauk SSSR 151 (6) (1963), 1264–1267 (transl. in: Soviet Math. Dokl. 4 (1963), 1169–1172).

    Google Scholar 

  14. Ju. L. Daleckii & M. G. Krein, “Stability of solutions of differential equation in Banach space”, Moscow, 1970 (transl. in: Transl. of Math. Monographs 43, Amer. Math. Soc., Providence, 1974).

  15. L. de Simon & G. Torelli, Linear second order differential equations with discontinuous coefficients in Hilbert spaces, Ann. Sc. Norm. Sup. Pisa, serie IV, 1 (1974), 131–154.

    Google Scholar 

  16. R. E. Edwards, “Functional analysis”, Holt, Rinehart & Winston, New York, 1965.

    Google Scholar 

  17. A. M. Fink, “Almost periodic differential equations”, Lecture Notes in Math. 377, Springer, Berlin-Heidelberg-New York 1974.

    Google Scholar 

  18. M. Fréchet, Les fonctions asymptotiquement presque-périodiques, Revue Scientifique, 79 (1941), 341–354.

    Google Scholar 

  19. G. Gilardi, Teoremi di regolarità per la soluzione di un'equazione differenziale astratta lineare del secondo ordine, Rend. Ist. Lombardo, Classe Sc. A 106 (1972), 641–675.

    Google Scholar 

  20. J. A. Goldstein, Time dependent hyperbolic equations, J. Functional Analysis. 4 (1969), 31–49.

    Google Scholar 

  21. A. E. Hurd & D. H. Sattinger, Question of existence and uniqueness for hyperbolic equations with discontinuous coefficients, Trans. Amer. Mat. Soc. 132 (1968), 159–174.

    Google Scholar 

  22. S. J. Jakubov, The Cauchy problem for second-order differential equations in Banach space, Dokl. Akad. Nauk SSSR 168 (1966), 759–762 (transl. in Soviet Math. Dokl. 7 (1966), 735–739).

    Google Scholar 

  23. S. J. Jakubov, Solvability of the Cauchy problem for abstract quasilinear hyperbolic equations of second order and their applications, Trudy Moskov. Mat. Obsc. 23 (1970), 37–60 (transl. in: Trans. Moscow Math. Soc. 23 (1970), 36–59).

    Google Scholar 

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

  25. S. G. Krein, “Linear differential equations in Banach space”, Moscow, 1967 (transl. in: Transl. of Math. Monographs, 29, Amer. Math. Soc., Providence, 1971).

  26. O. A. Ladyzenskaja, “The mixed problem for hyperbolic equations” (Russian) Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1953.

    Google Scholar 

  27. O. A. Ladyzenskaja & M. I. Višik, Boundary value problems for partial differential equations and certain classe of operator equations, Uspehi Mat. Nauk 11, n. 6 (1956), 41–97 (transl. in: Amer. Math. Soc. Transl. (2) 10 (1958), 223–281).

    Google Scholar 

  28. J. L. Lions, “Equation différentielles opérationnelles et problèmes aux limites”, Grundlehren 111, Springer, Berlin-Heidelberg-New York 1961.

    Google Scholar 

  29. J. L. Lions & E. Magenes, “Problèmes aux limites non homlogènes et applications”, Dunod, Paris, 1968.

    Google Scholar 

  30. L. A. Medeiros, The initial value problem for nonlinear wave equations in Hilbert space, Trans. Amer. Math. Soc. 136 (1969), 305–327.

    Google Scholar 

  31. V. A. Pogorelenko & P. E. Sobolevskii, Hyperbolic equations in Hilbert space, Sibirsk Mat. Ž. 8 (1967), 123–145 (transl. in: Siberian Math. J. 8 (1967–1), 92–108).

    Google Scholar 

  32. Z. Świętochowski, On second order Cauchy's problem in a Hilbert space with applications to the mixed problems for hyperbolic equations, I, II Ann. Univ. Mariae Curie-Skłodowska Sect. A 29 (1975), 123–144 (1977).

    Google Scholar 

  33. G. Torelli, Un complemento ad un teorema di J. L. Lions sulle equazioni differenziali astratte del secondo ordine, Rend. Sem. Mat. Padova, 34 (1964), 224–241.

    Google Scholar 

  34. F. Treves, Relations de domination entre opérateurs différentiels, Acta Math. 101 (1959), 1–139.

    Google Scholar 

  35. M. I. Višik, The Cauchy problem for equations with operator coefficients; mixed boundary value problems for systems of differential equations and approximation methods for their solution, Mat. Sbornik 39 (81) (1956), 51–148 (transl. in: Amer. Math. Soc. Transl. (2) 24 (1963), 173–278).

    Google Scholar 

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  1. Dipartimento di Matematica, Universitá di Pisa, Italy

    Alberto Arosio

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Communicated by J. L. Lions

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Arosio, A. Linear second order differential equations in hilbert spaces. Arch. Rational Mech. Anal. 86, 147–180 (1984). https://doi.org/10.1007/BF00275732

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