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Linear parabolic equations in Banach spaces with variable domains but constant interpolation spaces. (English) Zbl 0612.34057
The linear Cauchy problem in a Banach space E \[ (1)\quad u'(t)- A(t)u(t)=f(t),\quad t\in [0,T];\quad u(0)=x \] is studied; for each t, A(t) is assumed to be closed with time-dependent and possibly not dense domain \(D_{A(t)}\), and to generate an analytic semigroup. It is considered here an intermediate situation, frequently occurring in the applications (examples are given at the end of the paper) between the constant-domain case and the case of totally variable domains [see the authors’ papers Ann. Mat. Pura Appl., IV. Ser. 140, 1-55 (1985; Zbl 0579.34001), J. Funct. Anal. 60, 168-210 (1985; Zbl 0563.47028), J. Mat. Anal. Appl. 99, 9-64 (1984; Zbl 0555.34051)]: namely, the key hypothesis here is that for some \(\rho\in]0,1[\) the real interpolation space \(D_{A(t)}(\rho,\infty)=(D_{A(t)},E)_{1-\rho,\infty}\) is independent of t and that \(t\mapsto A(t)^{-1}\) is \(\alpha\)-Hölder continuous from E into \(D_{A(0)}(\rho,\infty)\) where \(\alpha\in [1-\rho,1[\). Results of existence, uniqueness and maximal regularity of the strict solution of (1) are proved when \(x\in D_{A(0)}\) and f is Hölder continuous in E or is bounded in \(D_{A(0)}(\rho,\infty)\) and continuous in E, provided suitable necessary and sufficient compatibility conditions holds. The proof consists firstly in establishing in ”a priori” representation formula for the solution, and secondly in showing, by a direct study of such formula, that it defines a function which is indeed a strict solution of (1). More general resuls than in this paper have been announced by the authors in the meanwhile [C. R. Acad. Sci. Paris, Sér. I 301, 107-110 (1985; Zbl 0581.34047)].

MSC:
34G10 Linear differential equations in abstract spaces
46M35 Abstract interpolation of topological vector spaces
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