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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),t[0,T];u(0)=u 0

when f: [0,T]E (a Banach space), u 0 E and A: D(A)EE 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.

47D03(Semi)groups of linear operators
47F05Partial differential operators
35K25Higher order parabolic equations, general
46K15Hilbert algebras