Characterizations of stabilizable sets for some parabolic equations in [formula omitted].

We consider the parabolic type equation in R n : (0.1) (∂ t + H) y (t , x) = 0 , (t , x) ∈ (0 , ∞) × R n ; y (0 , x) ∈ L 2 (R n) , where H can be one of the following operators: (i) a shifted fractional Laplacian; (i i) a shifted Hermite operator; (i i i) the Schrödinger operator with some general potentials. We call a subset E ⊂ R n as a stabilizable set for (0.1) , if there is a linear bounded operator K on L 2 (R n) so that the semigroup { e − t (H − χ E K) } t ≥ 0 is exponentially stable. (Here, χ E denotes the characteristic function of E , which is treated as a linear operator on L 2 (R n).) This paper presents different geometric characterizations of the stabilizable sets for (0.1) with different H. In particular, when H is a shifted fractional Laplacian, E ⊂ R n is a stabilizable set for (0.1) if and only if E ⊂ R n is a thick set, while when H is a shifted Hermite operator, E ⊂ R n is a stabilizable set for (0.1) if and only if E ⊂ R n is a set of positive measure. Our results, together with the results on the observable sets for (0.1) obtained in [1,19,25,33] , reveal such phenomena: for some H , the class of stabilizable sets contains strictly the class of observable sets, while for some other H , the classes of stabilizable sets and observable sets coincide. Besides, this paper gives some sufficient conditions on the stabilizable sets for (0.1) where H is the Schrödinger operator with some general potentials. [ABSTRACT FROM AUTHOR]