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It is known that a differential equation ẋ(t) = Ax(t) on the Banach space X (we assume the well-posedness, i.e., A generates a C0-semigroup {T(t)}t ≥ 0) is hyperbolic if X can be decomposed as X = X1 ⊕ X2 so that solutions ẋ(∙) starting from X1 (respectively, from X2) decay exponentially in forward time (respectively, in backward time). Hyperbolicity forces the solutions that start from X2 to exist...
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