Abstract
We show Carleman estimates for parabolic problems in divergence or non divergence form with degeneracy at the boundary or in the interior of the space domain. By them we obtain observability inequalities, proving that the problems are null controllable.
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Actually, in [29] much more general assumptions are assumed; precisely, the condition \(xa'(x)\le K_a a(x)\) is assumed only in a left neighborhood of \(x=0\). Thus, one chooses \(R>0\) such that \(2-\frac{xa'(x)}{a(x)}+4Rx^2\ge 2-K_a\) and takes \(\displaystyle p(x):=\int _0^x\frac{y}{a(y)}e^{Ry^2}dy\), see [29, p. 601].
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Fragnelli, G., Mugnai, D. (2021). The Non Singular Case: \(\lambda =0\). In: Control of Degenerate and Singular Parabolic Equations. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-69349-7_2
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DOI: https://doi.org/10.1007/978-3-030-69349-7_2
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