Bull Braz Math Soc, New Series DOI 10.1007/s00574-016-0001-0 Local Null Controllability of a Free-Boundary Problem for the Semilinear 1D Heat Equation Enrique Fernández-Cara 1 · Ivaldo Tributino de Sousa 2 Received: 6 May 2016 / Accepted: 2 June 2016 © Sociedade Brasileira de Matemática 2016 Abstract This paper deals with the local null control of a free-boundary problem for the 1D semilinear heat equation with distributed controls (locally supported in space) or boundary controls (acting at x = 0). In the main result we prove that, if the final time T is fixed and the initial state is sufficiently small, there exists controls that drive the state exactly to rest at time t = T . Keywords Null controllability · Free-boundary problems · 1D semilinear heat equation · Carleman estimates 1 Introduction Let T > 0 be given and let us assume that f : R → R is a globally Lipschitz continuous function. For any function L ∈ C 1 ([0, T ]) with 0 < L ∗ ≤ L (t ) ≤ B, t ∈ (0, T ), (1.1) Enrique Fernández-Cara is partially supported by Grant MTM2013-41286-P (DGI-MINECO, Spain) and CAPES (Brazil). B Ivaldo Tributino de Sousa tributino_1983@hotmail.com Enrique Fernández-Cara cara@us.es 1 Dpto. E.D.A.N., Univ. Sevilla, Aptdo. 1160, 41080 Seville, Spain 2 Dpto. Matemática, Univ. Federal do Ceará, Fortaleza, CE, Brazil 123