Null and approximate controllability for weakly blowing up semilinear heatequations

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 f(y) is such that /f(s)/ grows slower than /s/ log(3/2)(1 + /s/) as /s/ --> infinity. For in stance, this condition is fulfilled by any function f growing at infinity l ike /s/ log(P)(1 + /s/) with 1 < P < 3/2 (in this case, in the absence of c ontrol, blow-up occurs). We also prove that, for some functions f that beha ve at infinite like /s/ log(P)(1 + /s/) with P > 2, null controllability do es not hold. The problem remains open when f behaves at infinity like /s/ l og(P)(1 + /s/), with 3/2 less than or equal to p less than or equal to 2, R esults of the same kind are proved in the context of approximate controllab ility. (C) 2000 Editions scientifiques et medicales Elsevier SAS. AMS class ification: 93B05, 93C20.