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Some existence and regularity results for abstract non-autonomous parabolic equations. (English) Zbl 0555.34051
This paper contains the proofs, with examples and further remarks, of the results announced by the authors [Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 72, 322-329 (1982; Zbl 0525.34048)].

MSC:
34G10 Linear differential equations in abstract spaces
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[1] Agmon, S., On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems, Comm. pure appl. math., 15, 119-147, (1962) · Zbl 0109.32701
[2] Agmon, S.; Douglis, A.; Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, Comm. pure appl. math., 12, 623-727, (1959) · Zbl 0093.10401
[3] Baillon, J.B., Caractère borné de certains générateurs de semigroupes linéaires dans LES espaces de Banach, C. R. acad. sci. Paris Sér. A, 290, 757-760, (1980) · Zbl 0436.47027
[4] Butzer, P.L.; Berens, H., Semigroups of operators and approximation, (1967), Springer-Verlag New York/Berlin · Zbl 0164.43702
[5] Crandall, M.G.; Pazy, A., On the differentiability of weak solutions of a differential equation in Banach space, J. math. mech., 18, 1007-1016, (1969) · Zbl 0177.42901
[6] Da Prato, G.; Grisvard, P., Sommes d’opérateurs linéaires et équations différentielles opérationnelles, J. math. pures appl., 54, 305-387, (1975) · Zbl 0315.47009
[7] Da Prato, G.; Grisvard, P., Équations d’évolution abstraites non linéaires de type parabolyque, Ann. mat. pura appl., 120, 329-396, (1979), (4) · Zbl 0471.35036
[8] Da Prato, G.; Sinestrari, E., Hölder regularity for non-autonomous abstract parabolic equations, Israel J. math., 42, 1-19, (1982) · Zbl 0495.47031
[9] Dorroh, J.R., A linear evolution equation without a common dense core for the generators, J. differential equations, 31, 109-116, (1977) · Zbl 0397.34005
[10] Goldstein, J.A., On the absence of necessary conditions for linear evolution operators, (), 77-80 · Zbl 0332.47023
[11] Kato, T., On linear differential equations in Banach spaces, Comm. pure appl. math., 9, 479-486, (1956) · Zbl 0070.34602
[12] Kato, T., Abstract evolution equations of parabolic type in Banach and Hilbert spaces, Nagoya math. J., 19, 93-125, (1961) · Zbl 0114.06102
[13] Kato, T., Semigroups and temporally inhomogeneous evolution equations, (1963), C.I.M.E. 1 ° ciclo Varenna
[14] Kato, T., Perturbation theory for nonlinear operators, (1966), Springer-Verlag New York/Berlin
[15] Kato, T.; Tanabe, H., On the abstract evolution equations, Osaka math. J., 14, 107-133, (1962) · Zbl 0106.09302
[16] Krein, S.G., Linear differential equations in Banach spaces, Transl. math. monographs amer. math. soc., (1971), Providence, R. I. · Zbl 0636.34056
[17] Lions, J.L., Un théorème de traces; applications, C. R. acad. sci. Paris, 249, 2259-2261, (1959) · Zbl 0097.09601
[18] Lions, J.L., Équations différentielles opérationnelles et problèmes aux limites, (1961), Springer-Verlag New York/Berlin · Zbl 0098.31101
[19] Lions, J.L., Équations différentielles opérationnelles dans LES espaces de Hilbert, (1963), C.I.M.E. 1 ° ciclo Varenna · Zbl 0178.50704
[20] Lions, J.L.; Peetre, J., Sur une classe d’espaces d’interpolation, Inst. hautes études sci. publ. math., 19, 5-68, (1964) · Zbl 0148.11403
[21] Morrey, C.B., Multiple integrals in the calculus of variations, (1966), Springer-Verlag New York/Berlin · Zbl 0142.38701
[22] Pazy, A., Semigroups of linear operators and applications to partial differential equations, () · Zbl 0516.47023
[23] Peetre, J., Sur le nombre de paramètres dans la définition de certains espaces d’interpolation, Ricerche mat., 12, 248-261, (1963) · Zbl 0125.06501
[24] Poulsen, E.T., Evolutionsgleichungen in Banach Räumen, Math. Z., 90, 289-309, (1965) · Zbl 0141.13102
[25] Sinestrari, E., On the solutions of the inhomogeneous evolution equations in Banach spaces, Atti accad. naz. lincei rend. cl. sci. fis. mat. natur., 70, 12-17, (1981) · Zbl 0507.47027
[26] Sinestrari, E., Abstract semilinear equations in Banach spaces, Atti accad. naz. lincei rend. cl. sci. fis. mat. natur., 70, 81-86, (1981) · Zbl 0507.47026
[27] Sobolevski, P.E., First order differential equations in Hilbert space with a variable positive definite selfadjoint operator whose fractional power has a constant domain of definition (in Russian), Dokl. akad. nauk, 123, 984-987, (1958) · Zbl 0086.32101
[28] Sobolevski, P.E.; Sobolevski, P.E., Parabolic equations in a Banach space with an unbounded variable operator, a fractional power of which has a constant domain of definition (in Russian), Dokl. akad. nauk, Soviet math. dokl., 2, 545-548, (1961), (English transl.) · Zbl 0104.09303
[29] Stewart, H.B., Generation of analytic semigroups by strongly elliptic operators, Trans. amer. math. soc., 199, 141-162, (1974) · Zbl 0264.35043
[30] Stewart, H.B., Generation of analytic semigroups by strongly elliptic operators under general boundary conditions, Trans. amer. math. soc., 259, 299-310, (1980) · Zbl 0451.35033
[31] Tanabe, H., Remarks on the equations of evolution in a Banach space, Osaka math. J., 12, 145-166, (1960) · Zbl 0098.31202
[32] Tanabe, H., Note on singular perturbations for abstract differential equations, Osaka J. math., 1, 239-252, (1964) · Zbl 0135.37101
[33] Tanabe, H., Equations of evolution, (1979), Pitman London
[34] Tanabe, H.; Watanabe, M., Note on perturbation and degeneration of abstract differential equations in Banach space, Funcialaj ekvacioj, 9, 163-170, (1966) · Zbl 0186.47203
[35] Travis, C.C., Differentiability of weak solutions to an abstract inhomogeneous differential equation, (), 425-430 · Zbl 0484.34044
[36] Yagi, A., On the abstract evolution equations in Banach spaces, J. math. soc. Japan, 28, 290-303, (1976) · Zbl 0318.34068
[37] Yagi, A., On the abstract evolution equations of parabolic type, Osaka J. math., 14, 557-568, (1977) · Zbl 0371.47037
[38] Yosida, K., Functional analysis, (1968), Springer-Verlag New York/Berlin · Zbl 0217.16001
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