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On the abstract nonautonomous parabolic Cauchy problem in the case of constant domains. (English) Zbl 0579.34001
Let E be a Banach space, \(\{A(t)\}_{t\in [0,T]}^ a \)family of closed linear operators on E, C([0,T],E) the space of continuous functions [0,T]\(\to E\), and consider the linear non-autonomous Cauchy problem (P): \(u'(t)-A(t)u(t)=f(t),\) \(u(0)=x\), with \(x\in E\) and \(f\in C([0,T],E)\) prescribed. The authors study the case where \(\{\) A(t)\(\}\) is a family of infinitesimal generators of analytic semi-groups (parabolic case) whose domain does not depend on t but can be not dense in E (so generalizing results of previous authors who have considered the case D(A(t))\(\equiv D(A(0))\) dense in E). They obtain (heuristically) a representation integral formula for a solution of (P),and introduce several adequate Banach spaces of functions where they study in detail the operators and functions appearing in this formula.
Then, they study at some length, what happens (w.r.t. uniqueness, existence, regularity) with strict, classical and strong solutions of the problem under some hypotheses (some of the generalizing results already known). Finally, the theory is applied to classical non-autonomous boundary-initial value problems.
Reviewer: F.R.Dias Agudo

34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34G10 Linear differential equations in abstract spaces
47B38 Linear operators on function spaces (general)
Full Text: DOI
[1] Acquistapace, P.; Terreni, B., Some existence and regularity results for abstract nonautonomous parabolic equations, J. Math. Anal. Appl., 99, 9-64 (1984) · Zbl 0555.34051
[2] Agmon, S., On the eigenfunctions and the eigenvalues of general elliptic boundary value problems, Comm. Pure Appl. Math., 15, 119-147 (1962) · Zbl 0109.32701
[3] Amann, H.; Cesari, L.; Kannan, R.; Weinberger, H. F., Periodic solutions of semi-linear parabolic equations, Nonlinear analysis, a collection of papers in honour of E. H. Rothe (1978), New York: Academic Press, New York
[4] Baillon, J. B., Caractére borné de certains générateurs de semi-groupes linéaires dans les espaces de Banach, C. R. Acad. Sci. Paris, 290, 757-760 (1980) · Zbl 0436.47027
[5] Butzer, P. L.; Berens, H., Semi-groups of operators and approximation (1967), Berlin: Springer-Verlag, Berlin · Zbl 0164.43702
[6] Campanato, S., Equazioni ellittiche del secondo ordine e spazi L^2, λ, Ann. Mat. Pura Appl., (4), 69, 321-382 (1965) · Zbl 0145.36603
[7] Campanato, S., itEquazioni ellittiche non variazionali a coefficienti continui, Ann. Mat. Pura Appl., (4), 86, 125-154 (1970) · Zbl 0204.11701
[8] S.Campanato,Sistemi ellittici in forma divergenza. Regolarità all’interno, Quaderni Scuola Norm. Sup. Pisa, 1980. · Zbl 0453.35026
[9] Campanato, S., Generation of analytic semi-groups by elliptic operators of second order in Hölder spaces, Ann. Scuola Norm. Sup. Pisa, Cl. Scienze, (4), 8, 495-512 (1981) · Zbl 0475.35039
[10] 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
[11] Da Prato, G.; Grisvard, P., Sommes d’opérateurs linéaires et équations différentielles opérationnelles, J. Math. Pures Appl., 54, 305-387 (1975) · Zbl 0315.47009
[12] Da Prato, G.; Grisvard, P., Équations d’evolution abstraites non linéaires de type parabolyque, Ann. Mat. Pura Appl., (4), 120, 329-396 (1979) · Zbl 0471.35036
[13] Da Prato, G.; Sinestrari, E., Hölder regularity for non-autonomous abstract parabolic equations, Israel J. Math., 42, 1-19 (1982) · Zbl 0495.47031
[14] Fujita, H.; Kato, T., On the Navier-Stokes initial value problem, I, Arch. Rat. Mech. Anal., 16, 269-313 (1964) · Zbl 0126.42301
[15] Hartman, P., Ordinary differential equations (1964), New York: John Wiley & Sons, New York · Zbl 0125.32102
[16] Hille, E.; Phillips, R. S., Functional analysis and semi-groups (1957), Providence: Amer. Math. Soc. Colloquium Publ., Providence
[17] Iannelli, M., On the Green function for abstract evolution equation, Boll. Un. Mat. Ital., (4), 6, 154-174 (1972) · Zbl 0261.34038
[18] Kato, T., Remarks on pseudo-resolvents and infinitesimal generators of semi-groups, Proc. Japan Acad., 35, 467-468 (1959) · Zbl 0095.10502
[19] Kato, T., Perturbation theory for nonlinear operators (1966), Berlin: Springer-Verlag, Berlin
[20] Krein, S. G., Linear differential equations in Banach spaces (1971), Providence: Transl. Math. Monographs Amer. Math. Soc., Providence
[21] Lions, J. L., Théorèmes de trace et d’interpolation, I, Ann. Scuola Norm. Sup. Pisa, ser. III, 13, 389-403 (1959) · Zbl 0097.09502
[22] Lions, J. L.; Peetre, J., Sur une classe d’espaces d’interpolation, Inst. Hautes Études Sc. Publ. Math., 19, 5-68 (1964) · Zbl 0148.11403
[23] A.Lunardi,Interpolation spaces between domains of elliptic operators and spaces of continuous functions with applications to nonlinear parabolic equations, preprint Scuola Norm. Sup. Pisa, (1982), to appear in Math. Nachr.
[24] Martin, R. H., Nonlinear operators and differential equations in Banach spaces (1976), New York: John Wiley & Sons, New York
[25] Morrey, C. B., Multiple integral problems in the calculus of variations and related topics, Univ. California Publ. in Math., new ser., 1, 1-130 (1943)
[26] Morselli, L., Sistemi ellittici non variazionali, graduate thesis (1979), Pisa: Univ., Pisa
[27] A.Pazy,Semi-groups of linear operators and applications to partial differential equations, Lecture Notes, n. 10, Univ. Maryland, 1974.
[28] Poulsen, E. T., Evolutionsgleichungen in Banach Räumen, Math. Z., 90, 289-309 (1965) · Zbl 0141.13102
[29] Sinestrari, E., On the solutions of the inhomogeneous evolution equations in Banach spaces, Atti Acc. Naz. Lincei Rend. Cl. Sc. Fis. Mat. Nat., 70, 12-17 (1981) · Zbl 0507.47027
[30] E.Sinestrari,On the abstract Cauchy problem of parabolic type in spaces of continuous functions, to appear in J. Math. Anal. Appl. · Zbl 0589.47042
[31] Sobolevskii, P. E., On equations of parabolic type in Banach space, Trudy Moscow Mat. Obsc., 10, 297-350 (1961)
[32] Tanabe, H., A class of the equations of evolution in a Banach space, Osaka Math. J., 11, 124-165 (1959)
[33] Tanabe, H., Remarks on the equations of evolution in a Banach space, Osaka Math. J., 12, 145-166 (1960) · Zbl 0098.31202
[34] Tanabe, H., On the equations of evolution in a Banach space, Osaka Math, J., 12, 363-376 (1960) · Zbl 0098.31301
[35] Tanabe, H., Equations of evolution (1979), London: Pitman, London
[36] Travis, C. C., Differentiability of weak solutions to an abstract inhomogeneous differential equation, Proc. Amer. Math. Soc., 28, 425-430 (1981) · Zbl 0484.34044
[37] K.Yosida,On a class of infinitesimal generators and the integration problem of evolution equations, Proceedings of the 4° Berkeley Symposium in Mathematical Statistics and Probability, vol. 2, Berkeley, (1961), pp. 623-633.
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