Controllability for blowing up semilinear parabolic equations

Abstract

Abstract 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 grows slower than |s|log3/2(1+|s|) at infinity. We also prove that, for some nonlinearities that behave at infinity like |s|logp(1+|s|) with p>2, null controllability does not hold. Results of the same kind are proved in the context of approximate controllability.