Null and approximate controllability for weakly blowing up semilinear heat equations

We consider the semilinear heat equation in a bounded domain of R-d, with control on a subdomain and homogeneous Dirichlet boundary conditions. We prove that the system is null-controllable at any time provided a globally defined and bounded trajectory exists and the nonlinear term f(y) is such that /f(s)/ grows slower than /s/ log(3/2)(1 + /s/) as /s/ --> infinity. For instance, this condition is fulfilled by any function f growing at infinity like /s/ log(P)(1 + /s/) with 1 < P < 3/2 (in this case, in the absence of control, blow-up occurs). We also prove that, for some functions f that behave at infinite like /s/ log(P)(1 + /s/) with P > 2, null controllability does not hold. The problem remains open when f behaves at infinity like /s/ log(P)(1 + /s/), with 3/2 less than or equal to p less than or equal to 2, Results of the same kind are proved in the context of approximate controllability. (C) 2000 Editions scientifiques et medicales Elsevier SAS. AMS classification: 93B05, 93C20.