Abstract
We give some fundamental definitions and some Hardy-type inequalities with boundary or interior degeneracy. We also show the equivalence between null controllability and observability inequality.
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Notes
- 1.
Indeed, by [98, Theorem 1.3.1, Remarks 1.3.1 and 1.2.4] we have that \(u'\in C([0,T];[V,X]_{\frac{3}{4}})\subset C([0,T];[V,X]_1)= C([0,T];X)\), where \([V,X]_\theta \) denotes the \(\theta \)-th intermediate space between V and X.
- 2.
In the cited theorem \(u_0\in [D({\mathcal A}),X]_{\frac{1}{2}}\), but since \(D({\mathcal A})\) is dense in X we have that \([D({\mathcal A}),X]_{\frac{1}{2}}=V\), see [98, Chap. 1, Eq. (2.42)].
- 3.
With the approach therein, a clear application of the unique continuation principle is presented.
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Fragnelli, G., Mugnai, D. (2021). Mathematical Tools and Preliminary Results. In: Control of Degenerate and Singular Parabolic Equations. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-69349-7_1
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DOI: https://doi.org/10.1007/978-3-030-69349-7_1
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