zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A unified approach to abstract linear nonautonomous parabolic equations. (English) Zbl 0646.34006
The paper can be considered as a review, in which - on the basis of certain assumptions - a unified approach to abstract linear nonautonomous parabolic equations is proposed. In particular, the linear parabolic Cauchy problem $$ (1)\quad u'(t)-A(t)u(t)=f(t),\quad t\in [0,T],\quad u'(0)=x $$ is studied in a Banach space E, with $x\in E$ and f:[0,T]$\to E$ as prescribed data. In (1) $\{$ A(t)$\}$ $t\in [0,T]$ is a family of closed linear operators in E which are generators of analytic semigroups, and whose domain $D\sb{A(t)}$ may change with t and be not dense in E. Existence, uniqueness and regularity results are illustrated. The paper consists of 7 sections, rich in definitions, lemmas, propositions and theorems. In the last section examples and comparisons with the available literature are discussed.
Reviewer: V.C.Boffi

MSC:
34A12Initial value problems for ODE, existence, uniqueness, etc. of solutions
WorldCat.org
Full Text: Numdam EuDML
References:
[1] P. Acquistapace - B. TERRENI, Some existence and regularity results for abstract nonautonomous parabolic equations , J. Math. Anal. Appl. , 99 ( 1984 ), pp. 9 - 64 . MR 732703 | Zbl 0555.34051 · Zbl 0555.34051 · doi:10.1016/0022-247X(84)90234-8
[2] P. Acquistapace - B. TERRENI, On the abstract nonautonomous parabolic Cauchy problem in the case of constant domains , Ann. Mat. Pura Appl. , ( 4 ) 140 ( 1985 ), pp. 1 - 55 . MR 807632 | Zbl 0579.34001 · Zbl 0579.34001 · doi:10.1007/BF01776844
[3] P. Acquistapace - B. TERRENI, Maximal space regularity for abstract nonautonomous parabolic equations , J. Funct. Anal. , 60 ( 1985 ), pp. 168 - 210 . MR 777236 | Zbl 0563.47028 · Zbl 0563.47028 · doi:10.1016/0022-1236(85)90050-3
[4] P. Acquistapace - B. TERRENI, Existence and sharp regularity results for linear abstract parabolic integro-differential equations , Israel J. Math. 53 ( 1986 ), pp. 257 - 303 . MR 852481 | Zbl 0603.45019 · Zbl 0603.45019 · doi:10.1007/BF02786562
[5] P. Acquistapace - B. TERRENI, Linear parabolic equations in Banach spaces with variable domains but constant interpolation spaces , Ann. Sc. Norm. Sup. Pisa ( 4 ) 13 ( 1986 ), pp. 75 - 107 . Numdam | MR 863636 | Zbl 0612.34057 · Zbl 0612.34057 · numdam:ASNSP_1986_4_13_1_75_0 · eudml:83975
[6] P. Acquistapace - B. TERRENI, Une méthode unifiée pour l’étude des équations linéaires non autonomes paraboliques dans les espaces de Banach , C. R. Acad. Sci. Paris , 301 ( 1985 ), pp. 107 - 110 . MR 799604 | Zbl 0581.34047 · Zbl 0581.34047
[7] P.L. Butzer - H. Berens , Semigroups of Operators and Approximation , Springer-Verlag , Berlin / Heidelberg / New York , 1967 . MR 230022 | Zbl 0164.43702 · Zbl 0164.43702
[8] G. Da Prato - P. Grisvard , Sommes d’opérateurs linéaires et équations différentielles opérationnelles , J. Math. Pures Appl. , 54 ( 1975 ), pp. 305 - 387 . MR 442749 | Zbl 0315.47009 · Zbl 0315.47009
[9] G. Da Prato - P. Grisvard , Équations d’evolution abstraites non linéaires de type parabolique , Ann. Mat. Pura Appl. , ( 4 ) 120 ( 1979 ), pp. 329 - 396 . MR 551075 | Zbl 0471.35036 · Zbl 0471.35036 · doi:10.1007/BF02411952
[10] T. Kato , Remarks on pseudo-resolvents and infinitesimal generators of semigroups , Proc. Japan Acad. , 35 ( 1959 ), pp. 467 - 468 . Article | MR 117570 | Zbl 0095.10502 · Zbl 0095.10502 · doi:10.3792/pja/1195524254 · http://minidml.mathdoc.fr/cgi-bin/location?id=00257389
[11] T. Kato , Abstract evolution equations of parabolic type in Banach and Hilbert spaces , Nagoya Math. J. , 19 ( 1961 ), pp. 93 - 125 . Article | MR 143065 | Zbl 0114.06102 · Zbl 0114.06102 · http://minidml.mathdoc.fr/cgi-bin/location?id=00079331
[12] T. Kato , Semigroups and temporally inhomogeneous evolution equations , C.I.M.E. , 1^\circ ciclo, Varenna , 1963 . Zbl 0192.23902 · Zbl 0192.23902
[13] T. Kato - H. TANABE, On the abstract evolution equations , Osaka Math. J. , 14 ( 1962 ), pp. 107 - 133 . MR 140954 | Zbl 0106.09302 · Zbl 0106.09302
[14] R. Labbas - B. Terreni , Sommes d’opérateurs de type parabolique et elliptique , C. R. Acad. Sci. Paris , 301 ( 1985 ), pp. 169 - 172 . MR 801954 | Zbl 0601.35107 · Zbl 0601.35107
[15] J.L. Lions - J. Peetre , Sur une classe d’espaces d’interpolation , Inst. Hautes Études Sci. Publ. Math. , 19 ( 1964 ), pp. 5 - 68 . Numdam | MR 165343 | Zbl 0148.11403 · Zbl 0148.11403 · doi:10.1007/BF02684796 · numdam:PMIHES_1964__19__5_0 · eudml:103841
[16] E. Sinestrari , On the abstract Cauchy problem of parabolic type in spaces of continuous functions , J. Math. Anal. Appl. , 107 ( 1985 ), pp. 16 - 66 . MR 786012 | Zbl 0589.47042 · Zbl 0589.47042 · doi:10.1016/0022-247X(85)90353-1
[17] P.E. Sobolevskii , On equations of parabolic type in Banach space , Trudy Moscow Mat. Obsc. , 10 ( 1961 ), pp. 297 - 350 (in Russian); English transl.: Amer. Math. Soc. Transl. , 49 ( 1965 ), pp. 1 - 62 . MR 141900
[18] H. Tanabe , On the equations of evolution in a Banach space , Osaka Math. J. , 12 ( 1960 ), pp. 363 - 376 . MR 125455 | Zbl 0098.31301 · Zbl 0098.31301
[19] H. Tanabe , Note on singular perturbation for abstract differential equations , Osaka J. Math. , 1 ( 1964 ), pp. 239 - 252 . MR 172142 | Zbl 0135.37101 · Zbl 0135.37101
[20] H. Triebel , Interpolation Theory, Function Spaces, Differential Operators , North-Holland , Amsterdam / New York / Oxford , 1978 . MR 503903 | Zbl 0387.46032 · Zbl 0387.46032
[21] A. Yagi , On the abstract linear evolution equations in Banach spaces , J. Math. Soc. Japan , 28 ( 1976 ), pp. 290 - 303 . Article | MR 397478 | Zbl 0318.34068 · Zbl 0318.34068 · doi:10.2969/jmsj/02820290 · http://minidml.mathdoc.fr/cgi-bin/location?id=00317028
[22] A. Yagi , On the abstract evolution equation of parabolic type , Osaka J. Math. , 14 ( 1977 ), pp. 557 - 568 . MR 499587 | Zbl 0371.47037 · Zbl 0371.47037