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Null and approximate controllability for weakly blowing up semilinear heat equations. (English) Zbl 0970.93023
Summary: 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 $f\left(y\right)$ is such that $|f\left(s\right)|$ grows slower than $|s|{log}^{3/2}\left(1+|s|\right)$ as $|s|\to \infty$. For instance, this condition is fulfilled by any function $f$ growing at infinity like $|s|{log}^{p}\left(1+|s|\right)$ with $1 (in this case, in the absence of control, blow-up occurs). We also prove that, for some functions $f$ that behave at infinity like $|s|{log}^{p}\left(1+|s|\right)$ with $p>2$, null controllability does not hold. The problem remains open when $f$ behaves at infinity like $|s|{log}^{p}\left(1+|s|\right)$, with $3/2\le p\le 2$. Results of the same kind are proved in the case of approximate controllability.

##### MSC:
 93C20 Control systems governed by PDE 93B05 Controllability 93C10 Nonlinear control systems 35B37 PDE in connection with control problems (MSC2000)
##### Keywords:
semilinear heat equation; controllability