In their seminal 1985 paper Giga and Kohn analysed the blow-up behaviour of the sub-critical Fujita equation via a Liouville theorem for an associated elliptic equation of Ornstein-Uhlenbeck type. In a subsequent work Giga provided a conditional extension of this Liouville theorem to a natural broader class of semilinear Ornstein-Uhlenbeck equations and posed a question of its unconditional validity, i.e. in the class of bounded entire solutions. In this talk I will show how a careful use of a generalized Rellich-Pohozaev type argument provides a positive answer to this question.