Optimal Liouville theorem for a semilinear Ornstein-Uhlenbeck equation

In their seminal 1985 paper, Giga and Kohn analyzed the blow-up behavior of the subcritical Fujita equation via a Liouville theorem for an associated elliptic equation of Ornstein-Uhlenbeck type. In 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 careful use of a generalized Rellich-Pohozaev type argument provides a positive answer to this question and point to some interesting byproducts of this technique.