We consider a semilinear heat equation in an unbounded domain Ω with partially known initial data. The insensitizing problem consists in finding a control function such that some functional of the state is locally insensitive to the perturbations of these initial data. For bounded domains Bodart and Fabre proved the existence of insensitizing controls of the norm of the observation of the solution in an open subset of the domain. In this paper we prove similar results when Ω is unbounded. We consider the problem in bounded domains of the form Ωr = Ω ∩ Br where Br denotes the ball centered in zero of radius r We show that for r large enough the control proposed by Bodart and Fabre for the problem in Ωr, provides an insensitizing control for our problem in Ω.