Controllability for blowing up semilinear parabolic equations

We consider the semilinear heat equation in a bounded domain of R-d, with c ontrol on a subdomain and homogeneous Dirichlet boundary conditions. We pro ve that the system is null-controllable at any time provided a globally def ined and bounded trajectory exists and the nonlinear term grows slower than \s\log(3/2)(1 + \s\) at infinity. We also prove that, for some nonlinearit ies that behave at infinity like \s\log(p)(1 + \s\) with p > 2, null contro llability does not hold. Results of the same kind are proved in the context of approximate controllability. (C) 2000 Academie des sciences/Editions sc ientifiques et medicales Elsevier SAS.