Regional controllability of semilinear parabolic equations in unbounded domains

It is known that the null controllability property for parabolic equations in unbounded domains takes weaker forms than the same property in bounded domains. In particular, as shown by S. Micu and E. Zuazua [On the lack of null controllability of the heat equation on the half line, Trans. Amer. Math. Soc. 353 (2001), 1635-1659] the solution of a parabolic equation in an unbounded domain cannot be steered to zero by a control acting on a bounded set. In this paper, we study ”regional” null controllability properties of semilinear parabolic equations in unbounded domains, proving that we can drive the solution to rest in time T on a suitable subregion of the space domain, determined by the support of controls.