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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 $$\mathbb{R}^d$$, 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 $$f(y)$$ 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 $$3/2\leq p\leq 2$$. Results of the same kind are proved in the case of approximate controllability.

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