On the abstract Cauchy problem of parabolic type in spaces of continuous functions. (English) Zbl 0589.47042

The main object of this paper is the study of the problem \[ u'(t)=Au(t)+f(t),\quad t\in [0,T];\quad u(0)=u_ 0 \] when f: [0,T]\(\to E\) (a Banach space), \(u_ 0\in E\) and A: D(A)\(\subseteq E\to E\) verifies all the properties of the generators of analytic semigroups with the possible exception of the density of D(A) in E. This gives the possibility of applying the abstract theory to Cauchy-Dirichlet problems for parabolic equations and obtain maximal regularity results in spaces of Hölder continuous functions.


47D03 Groups and semigroups of linear operators
47F05 General theory of partial differential operators
35K25 Higher-order parabolic equations
46K15 Hilbert algebras
Full Text: DOI


[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] Ardito, A.; Ricciardi, P., Existence and regularity for linear delay partial differential equations, Nonlinear Anal., 4, 411-414 (1980) · Zbl 0433.35066
[3] Bers, L.; John, F.; Schechter, M., Partial Differential Equations (1964), Interscience: Interscience New York
[4] Berens, H.; Butzer, P. L., Approximation theorems for semi-group operators in intermediate spaces, Bull. Amer. Math. Soc. (N.S.), 70, 689-692 (1964) · Zbl 0202.42403
[5] Butzer, P. L.; Berens, H., Semigroups of Operators and Approximation (1967), Springer: Springer Berlin · Zbl 0164.43702
[6] Campanato, S., Generation of analytic semi-groups by elliptic operators of second order in Hölder spaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 8, 495-512 (1981) · Zbl 0475.35039
[7] Da Prato, G.; Grisvard, P., Sommes d’opérateurs linéaires et équations différentielles opérationnelles, J. Math. Pure Appl., 54, 305-387 (1975) · Zbl 0315.47009
[8] Da Prato, G.; Grisvard, P., Equations d’évolution abstraites non linéaires de type parabolique, Ann. Mat. Pura Appl. (4), 120, 329-396 (1979) · Zbl 0471.35036
[9] Da Prato, G.; Sinestrari, E., Hölder regularity for non-autonomous abstract parabolic equations, Israel J. Math., 42, 1-19 (1982) · Zbl 0495.47031
[10] Dunford, N.; Schwartz, J. T., Linear Operators I (1957), Interscience: Interscience New York
[11] Friedman, A., Partial Differential Equations (1969), Holt: Holt New York
[12] Kato, T., Perturbation Theory for Linear Operators (1966), Springer: Springer Berlin · Zbl 0148.12601
[13] Ladas, G. E.; Lakshmikantham, V., Differential Equations in Abstract Spaces (1972), Academic Press: Academic Press New York · Zbl 0257.34002
[14] Ladyzenskaja, O. A.; Solonnikov, V. A.; Uralceva, N. N., Linear and Quasilinear Equations of Parabolic Type (1968), American Mathematics Society
[15] Lions, J. L., Théorèmes de trace et d’interpolation (I), Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 13, 389-403 (1959) · Zbl 0097.09502
[16] Lunardi, A., Analyticity of the maximal solution of an abstract nonlinear parabolic equation, Nonlinear Anal, 6, 503-521 (1982) · Zbl 0486.35017
[17] A. LunardiMath. Nachr.; A. LunardiMath. Nachr. · Zbl 0568.47035
[18] Martin, R. H., Nonlinear Operators and Differential Equations in Banach Spaces (1976), Wiley: Wiley New York
[19] Pazy, A., Semi-groups of linear operators and applications to partial differential equations, (lecture notes (1974), University of Maryland) · Zbl 0516.47023
[20] Sinestrari, E., On the solutions of the inhomogeneous evolution equations in Banach spaces, Rend. Accad. Naz. Lincei, 70, 12-17 (1981) · Zbl 0507.47027
[21] Sinestrari, E., Continuous interpolation spaces and spatial regularity in nonlinear Volterra integrodifferential equations, J. Integral Equations, 5, 287-308 (1983) · Zbl 0519.45013
[22] Sinestrari, E., Classical solutions of parabolic equations in Hölder spaces, Rend. Accad. Naz. Lincei, 73, 289-297 (1983) · Zbl 0591.35011
[23] Sinestrari, E.; Vernole, P., Semilinear evolution equations in interpolation spaces, Nonlinear Anal., 1, 249-261 (1977) · Zbl 0357.34061
[24] Stewart, B., Generation of analytic semigroups by strongly elliptic operators, Trans. Amer. Math. Soc., 199, 141-162 (1974) · Zbl 0264.35043
[25] Stewart, B., Generation of analytic semigroups by strongly elliptic operators under general boundary conditions, Trans. Amer. Math. Soc., 259, 299-310 (1980) · Zbl 0451.35033
[26] Tanabe, H., Equations of Evolution (1979), Pitman: Pitman London
[27] Triebel, H., Interpolation Theory, Functions Spaces, Differential Operators (1978), North-Holland: North-Holland Amsterdam
[28] Von Wahl, W., Gebrochene Potenzen eines elliptischen Operators und parabolische Differentialgleichungen in Räumen hölderstetiger Funktionen, Nachr. Akad. Wiss. Göttingen, 11, 231-258 (1972) · Zbl 0251.35052
[29] Yosida, K., Functional Analysis (1968), Springer: Springer Berlin · Zbl 0217.16001
[30] Zygmund, A., Trigonometric Series (1959), Cambridge Univ. Press: Cambridge Univ. Press London/New York · JFM 58.0280.01
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