Null and approximate controllability for weakly blowing up semilinear heat equations

Abstract

We consider the semilinear heat equation in a bounded domain of ℝd, 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|log3/2(1+|s|) as |s|→∞ . For instance, this condition is fulfilled by any function f growing at infinity like |s|logp(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|logp(1+|s|) with p>2, null controllability does not hold. The problem remains open when f behaves at infinity like |s|logp(1+|s|), with 3/2≤p≤2 . Results of the same kind are proved in the context of approximate controllability.