A Sufficient Condition for Exponential Dichotomy of Parabolic Evolution Equations

In this note we establish the exponential dichotomy of the abstract Cauchy problem ( CP )     { d d t u ( t ) = A ( t ) u ( t ) ,                 t ≥ s ,             u ( s ) = x , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187810/28005645-324c-4d25-a9e4-00a49a74dffd/content/equ_chapter12_149_1.tif"/> assuming that A(t), t ∊ ℝ, generates an analytic semigroup (e ^τA(t))_τ≥0 and A(·)⁻¹satisfies a certain Hölder condition (see (P) below). Moreover, each semigroup (e ^τA(t))_τ≥0 is supposed to have exponential dichotomy and the Hölder constant of A(·)⁻¹ must be sufficiently small. Already simple matrix examples, [6, p.3], [15, Ex.3.4], show that one cannot omit this smallness condition.