Differential operators with non dense domain. (English) Zbl 0652.34069

da Prato, Guiseppe; Sinestrari, E.

Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 14, No. 2, 285-344 (1987).

The Cauchy problem in the Banach space E of the form \(u'(t)=Au(t)+f(t),\) \(t\in [0,T]\); \(u(0)=u_ 0\) where A:D\(\subseteq E\to E\) is a closed linear operator with non dense domain D, f: [0,T]\(\to E\), \(u_ 0\in E\), is considered. Analogues of well-known Hille-Yosida theorem (the assumption of the density of D in E is excepted) are proved. A more particular situation in which A generates an analytical semigroup (not necessarily strongly continuous at the origin) is investigated. As an application, the authors consider several examples of partial differential equations of hyperbolic and ultraparabolic type, and parabolic equations with infinitely many variables.

Reviewer: R.R.Akhmerov

MSC:

34G10	Linear differential equations in abstract spaces
47D03	Groups and semigroups of linear operators
35K70	Ultraparabolic equations, pseudoparabolic equations, etc.
35L20	Initial-boundary value problems for second-order hyperbolic equations
35K99	Parabolic equations and parabolic systems

Keywords:

first order differential equations; Cauchy problem; Banach space; linear operator with non dense domain

References:

[1]	D. Baillon . Caractère borné de certains générateurs de semigroupes linéaires dans les espaces de Banach . C.R. Acad. Sc. Paris 290 ( 1980 ), 757 - 760 . Zbl 0436.47027 · Zbl 0436.47027
[2]	G. Da Prato , P. Grisvard . Sommes d’ opérateurs linéaires et équations différentielles opérationnelles . J. Math. Pures Appl. 54 ( 1975 ), 305 - 387 . MR 442749 | Zbl 0315.47009 · Zbl 0315.47009
[3]	G. Da Prato . Some results on parabolic evolution equations with infinitely many variables . J. Differential Eq. 68 , 2 ( 1987 ), 281 - 297 . MR 892028 | Zbl 0628.35044 · Zbl 0628.35044
[4]	G. Di Blasio , M. Iannelli , E. Sinestrari . An abstract partial differential equation with a boundary condition of renewal type . Boll. U.M.I. An. Funz. 18 ( 1981 ), 259 - 274 . MR 631582 | Zbl 0474.45009 · Zbl 0474.45009
[5]	G. Di Blasio . Linear parabolic evolution equations in LP-spaces . Annali Mat. 138 ( 1984 ), 55 - 104 . MR 779538 | Zbl 0568.35047 · Zbl 0568.35047
[6]	A. Favini . Su una equazione astratta alle derivate parziali . Rend. Mat. 9 ( 1976 ), 665 - 700 . MR 433064 | Zbl 0342.35054 · Zbl 0342.35054
[7]	T. Kato . Remarks on pseudo-resolvents and infinitesimal generators of semi-groups . Proc. Japan Ac. 35 ( 1959 ), 467 - 468 . Article | MR 117570 | Zbl 0095.10502 · Zbl 0095.10502
[8]	T. Kato . Perturbation theory for linear operators . Springer , Berlin 1966 . Zbl 0148.12601 · Zbl 0148.12601
[9]	A. Luna Interpolation spaces between domains of elliptic operators and spaces of continuous functions with applications to nonlinear parabolic equations . Math. Nachr. 121 ( 1985 ), 295 - 318 . MR 809327 | Zbl 0568.47035 · Zbl 0568.47035
[10]	A. Pazy . Semigroups of linear operators and applications to partial differential equations . Springer , Berlin 1983 . MR 710486 | Zbl 0516.47023 · Zbl 0516.47023
[11]	E. Sinestrari . Contrnuous interpolation spaces and spatial regularity in nonlinear Volterra integrodifferential equations . J. Integral Equations 5 ( 1983 ), 287 - 308 . MR 714457 | Zbl 0519.45013 · Zbl 0519.45013
[12]	E. Sinestrari . On the abstract Cauchy problem of parabolic type in spaces of continuous functions . J. Math. Anal. Appl. 107 ( 1985 ), 16 - 66 . MR 786012 | Zbl 0589.47042 · Zbl 0589.47042
[13]	B. Stewart . Generation of analytic semigroups by strongly elliptic operators . Trans. Amer. Math. Soc. 199 ( 1974 ), 141 - 162 . MR 358067 | Zbl 0264.35043 · Zbl 0264.35043
[14]	W. Von Wahl . Gebrochene Potenzen eines elliptischen Operators und parabolische Differentialgleichungen in Räumen hölderstetiger Funktionen , Nachr. Akad. Wiss. Göttingen 11 ( 1972 ), 231 - 258 . MR 313636 | Zbl 0251.35052 · Zbl 0251.35052
[15]	K. Yosida . Functional Analysis . Springer , Berlin 1968 . · Zbl 0152.32102