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Maximal regularity for evolution equations by interpolation and extrapolation. (English) Zbl 0593.47041
Let A generate an analytic semigroup on a Banach space E. Suppose $$f\in C([0,T];E_ 1)$$. Does it follow that both u’ and Au belong to $$C([0,T];E_ 1)$$ with $$E_ 1=E?$$ The answer is ”no” in general. [Cf. J.-B. Baillon, C. R. Acad. Sci., Paris, Sér. A 290, 757-760 (1980; Zbl 0436.47027)]. But the answer is ”yes” if $$E_ 1$$ is a suitable interpolation space between D(A) and E; this was shown by the authors earlier [Ann. Mat. Pura Appl., IV. Ser. 120, 329-396 (1979; Zbl 0471.35036)]. When the answer is ”yes”, one speaks of the equation satisfying ”maximal regularity.” There are some technical complications in extending this result to the case when $$A=A(t)$$ is time dependent. The reason is analogous to the result that if $$A_ 1$$ and $$A_ 2$$ generate uniformly bounded semigroups on E and if $$D(A_ 1)=D(A_ 2)$$, then $$D((A_ 1)^{\alpha})=D((-A_ 2)^{\alpha})$$ holds for $$0<\alpha <1$$ but fails to hold for $$\alpha >1$$. The authors’ approach is to define extrapolation spaces, which are interpolation spaces between E and a superspace F of E. (F is defined with respect to a suitable reference operator $$A_ 0$$ as $$E\times E/G$$, where G is the graph of $$A_ 0.)$$ The authors indicate that they hope to extend their maximal regularity results to a nonlinear context in a future paper.
Reviewer: J.A.Goldstein

##### MSC:
 47D03 Groups and semigroups of linear operators 46M35 Abstract interpolation of topological vector spaces 34G10 Linear differential equations in abstract spaces
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##### References:
 [1] Baillon, Caractère borné de certains semi-groupes linéaires dans LES espaces de Banach, Note C. R. acad. sci. Paris ser. I math., 290, 28 IV, 757-760, (1980) · Zbl 0436.47027 [2] Da Prato, G; Grisvard, P, Équations d’évolution abstraites non linéaires de type parabolique, Ann. mat. pura appl., CXX, IV, 329-396, (1979) · Zbl 0471.35036 [3] Tanabe, Evolution equations of parabolic type, (), 610-613 · Zbl 0104.34002
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