Null controllability for a heat equation with a singular inverse-square potential involving the distance to the boundary function

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• 作者：Umberto Biccari^a ; ¹ ; ^{ubiccari@bcamath.org" class="auth_mail" title="E-mail the corresponding author} ; ^{u.biccari@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Enrique Zuazua^b ; ² ; ^{enrique.zuazua@uam.es" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35K05 ; 93B05 ; 93B07
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：5 September 2016
• 年：2016
• 卷：261
• 期：5
• 页码：2809-2853
• 全文大小：525 K

文摘

This article is devoted to the analysis of control properties for a heat equation with a singular potential μ/δ²$\mu /{\delta }^2$, defined on a bounded C²$C^2$ domain Ω⊂R^N$\mathrm{\Omega }\subset {\mathbb{R}}^N$, where δ   is the distance to the boundary function. More precisely, we show that for any μ≤1/4$\mu \le 1/4$ the system is exactly null controllable using a distributed control located in any open subset of Ω, while for μ>1/4$\mu >1/4$ there is no way of preventing the solutions of the equation from blowing-up. The result is obtained applying a new Carleman estimate.