Maximal regularity for evolution equations by interpolation and extrapolation
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. \textit{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.
scientific article; zbMATH DE number 3953638