  In this paper, we first prove a uniform upper bound on costs of null controls
for semilinear heat equations with globally Lipschitz nonlinearity on a
sequence of increasing domains, where the controls are acted on an
equidistributed set that spreads out in the whole Euclidean space R^N . As an
application, we then show the exactly null controllability for this semilinear
heat equation in R^N . The main novelty here is that the upper bound on costs
of null controls for such kind of equations in large but bounded domains can be
made uniformly with respect to the sizes of domains under consideration. The
latter is crucial when one uses a suitable approximation argument to derive the
global null controllability for the semilinear heat equation in R^N . This
allows us to overcome the well-known problem of the lack of compactness
embedding arising in the study of null controllability for nonlinear PDEs in
generally unbounded domains.
