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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 $\bbfR^d$, 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 $|s|\to\infty$. 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 infinity 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 $3/2\le p\le 2$. Results of the same kind are proved in the case of approximate controllability.

93C20Control systems governed by PDE
93C10Nonlinear control systems
35B37PDE in connection with control problems (MSC2000)
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