Abstract
We find a new construction of the evolution operatorG(t, s) associated to a family {A(t), 0≦t≦T} of generators of analytic semigroups in a Banach spaceX. We study the dependence ofG (t, s) ont ands, and we give regularity results for the solution of the i.v.p.u′(t)=A(t)u(t)+f(t),u(0)=x.
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P. Acquistapace and B. Terreni,On the abstract non-autonomous parabolic Cauchy problem in the case of constant domains, Ann. Mat. Pura Appl. (IV)140 (1985), 1–55.
P. Acquistapace and B. Terreni,Maximal space regularity for abstract linear non-autonomous parabolic equations, J. Funct. Anal.60 (1985), 168–210.
P. Acquistapace and B. Terreni,A unified approach to abstract linear non-autonomous parabolic equations, Rend. Sem. Mat. Univ. Padova (to appear).
P. L. Butzer and H. Berens,Semigroups of Operators and Approximation, Springer-Verlag, Berlin, 1967.
G. Da Prato and E. Sinestrari,Hölder regularity for non-autonomous abstract parabolic equations, Isr. J. Math.42 (1982), 1–19.
A. Lunardi,Bounded solutions of linear periodic abstract parabolic equations, preprint Dip. Mat. Univ. Pisa (1986).
A. Lunardi and E. Sinestrari, {ie314-1}-regularity for non-autonomous linear integrodifferential equations of parabolic type, J. Differ. Equ.63 (1986), 88–116.
E. Sinestrari,On the abstract Cauchy problem of parabolic type in spaces of continuous functions, J. Math. Anal. Appl.107 (1985), 16–66.
P. E. Sobolevskiī,Equations of parabolic type in a Banach space, Am. Math. Soc. Transl.49 (1966), 1–62.
P. E. Sobolevskiī,Coerciveness inequalities for abstract parabolic equations, Soviet Math.5 (1964), 894–897.
H. Tanabe,On the equations of evolution in a Banach space, Osaka J. Math.12 (1960), 363–376.
H. Triebel,Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978.
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Lunardi, A. On the evolution operator for abstract parabolic equations. Israel J. Math. 60, 281–314 (1987). https://doi.org/10.1007/BF02780395
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DOI: https://doi.org/10.1007/BF02780395