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On the abstract Cauchy problem of parabolic type in spaces of continuous functions. (English) Zbl 0589.47042
The main object of this paper is the study of the problem \[ u'(t)=Au(t)+f(t),\quad t\in [0,T];\quad u(0)=u_ 0 \] when f: [0,T]\(\to E\) (a Banach space), \(u_ 0\in E\) and A: D(A)\(\subseteq E\to E\) verifies all the properties of the generators of analytic semigroups with the possible exception of the density of D(A) in E. This gives the possibility of applying the abstract theory to Cauchy-Dirichlet problems for parabolic equations and obtain maximal regularity results in spaces of Hölder continuous functions.

MSC:
47D03 Groups and semigroups of linear operators
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
35K25 Higher-order parabolic equations
46K15 Hilbert algebras
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