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Approximate controllability of a semilinear heat equation in ${ℝ}^{N}$. (English) Zbl 0935.93011

This note treats the approximate controllability of the semilinear heat equation in unbounded domains ${\Omega }$ when the control acts on the interior of ${\Omega }$.

When ${\Omega }$ is a bounded set, C. Fabre, J.-P. Puel and E. Zuazua [IMA Vol. Math. Appl. 70, 73-91 (1995; Zbl 0822.35075)] proved the approximate controllability of the semilinear heat equation in ${L}^{p}\left({\Omega }\right)$, $1\le p<\infty$. Their proof is divided into two parts: a) approximate controllability of the linearized systems, b) a fixed point technique. This method cannot be applied when ${\Omega }$ is an unbounded set since the compactness of Sobolev’s embeddings is one of the main ingredients used in b). In a note, L. de Teresa and E. Zuazua [Nonlinear Anal. (to appear)] prove the approximate controllability of the semilinear heat equation in unbounded domains by an approximation method.

Also the control problem in bounded domains is studied. In this note previously introduced techniques are adapted to unbounded domains by introducing the weighted Sobolev spaces of M. Escobedo and O. Kavian [Nonlinear Anal., Theory Methods Appl. 11, 1103-1133 (1987; Zbl 0639.35038)] which permit (guarantee) the compactness of Sobolev embeddings.

##### MSC:
 93B05 Controllability 35K05 Heat equation 35K55 Nonlinear parabolic equations