Regional null controllability for degenerate heat equations

We are interested in a null controllability problem for a class of strongly degenerate heat equations. First for all T > 0, we prove a regional null controllability result at time T at least in the region where the equation is not degenerate. The proof is based on an adequate observability inequality for the homogeneous adjoint problem. This inequality is obtained by application of Carleman estimates combined with the introduction of cut-off functions. Then we improve this result: for all T' > T, we obtain a result of persistent regional null controllability during the time interval [T, T']. Finally we give similar results for the (non degenerate) heat equation in unbounded domain.