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Bounded solutions of parabolic equations in continuous function spaces. (English) Zbl 1151.34049
Summary: This paper is concerned with the existence of bounded mild solutions to equations of the form
$u^{\prime}(t) = Au(t) + f(t),$ where $$A$$ generates a holomorphic semigroup that is not necessarily strongly continuous, and $$f$$ is a bounded function. This problem arises when one considers a parabolic equation in spaces of continuous functions. The obtained results, that are stated in terms of spectral properties of the spectrum of $$A$$ and the uniform spectrum of $$f$$, extend previous ones.

##### MSC:
 34G10 Linear differential equations in abstract spaces 35K90 Abstract parabolic equations
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