Approximate controllability for the semilinear heat equation in RN involving gradient terms

We prove the approximate controllability of the semilinear heat equation in RN, when the nonlinear term is globally Lipschitz and depends both on the state u and its spatial gradient Ñu. The approximate controllability is viewed as the limit of a sequence of optimal control problems. In order to avoid the difficulties related to the lack of compactness of the Sobolev embeddings, we work with the similarity variables and use weighted Sobolev spaces