Null controllability in large time of a parabolic equation involving the Grushin operator with an inverse-square potential
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  • 作者：Cung The Anh ; Vu Manh Toi
  • 关键词：Null controllability ; Uniform observability ; Grushin operator ; Hardy inequality ; Carleman inequality ; Dissipation speed
  • 刊名：NoDEA : Nonlinear Differential Equations and Applications
  • 出版年：2016
  • 出版时间：April 2016
  • 年：2016
  • 卷：23
  • 期：2
  • 全文大小：679 KB
  • 参考文献：1.D’Ambrosio L.: Hardy inequalities related to Grushin type operators. Proc. Am. Math. Soc 132, 725–734 (2003)CrossRef MATH MathSciNet
    2.Alabau-Boussouira F., Cannarsa P., Fragnelli G.: Carleman estimates for weakly degenerate parabolic operators with applications to null controllability. J. Evol. Equ 6, 161–204 (2006)CrossRef MATH MathSciNet
    3.Anh C.T., Toi V.M.: Null controllability of a parabolic equation involving the Grushin operator in some multi-dimensional domains. Nonlinear Anal. 93, 181–196 (2013)CrossRef MATH MathSciNet
    4.Beauchard K., Cannarsa P., Guglielmi R.: Null controllability of Grushin-type operators in dimension two. J. Eur. Math. Soc 16, 67–101 (2014)CrossRef MATH MathSciNet
    5.Beauchard K., Cannarsa P., Yamamoto M.: Inverse source problem and null controllability for multidimensional parabolic operators of Grushin type. Inverse Probl. 30(2), 025006 (2014)CrossRef MATH MathSciNet
    6.Brezis H., Vázquez J.L.: Blowup solutions of some nonlinear elliptic problems. Rev. Mat. Univ. Complut. Madr. 10, 443–469 (1997)MATH
    7.Cannarsa P., Martinez P., Vancostenoble J.: Carleman estimates for a class of degenerate parabolic operators. SIAM J. Control Optim. 47, 1–19 (2008)CrossRef MATH MathSciNet
    8.Cannarsa P., Fragnelli G.: Null controllability of semilinear degenerate parabolic equations in bounded domains. EJDE 2006(136), 1–20 (2006)MATH MathSciNet
    9.Cannarsa P., Fragnelli G., Vancostenoble J.: Regional controllability of semilinear degenerate parabolic equations in bounded domains. J. Math. Anal. Appl 320, 804–818 (2006)CrossRef MATH MathSciNet
    10.Cannarsa P., Martinez P., Vancostenoble J.: Persistent regional controllability for a class of degenerate parabolic equations. Comm. Pure Appl. Anal. 3, 607–635 (2004)CrossRef MATH MathSciNet
    11.Cannarsa P., Martinez P., Vancostenoble J.: Null controllability of degenerate heat equations. Adv. Differ. Equ. 10, 153–190 (2005)MATH MathSciNet
    12.Cannarsa, P., Guglielmi, R.: Null controllability in large time for the parabolic Grushin operator with singular potential, geometric control theory and sub-riemannian geometry, Springer INdAM Series Volume 5, 87–102 (2014)
    13.Cazacu C.: Controllability of the heat equation with an inverse-square potential localized on the boundary. SIAM J. Control Optim. 52, 2055–2089 (2014)CrossRef MATH MathSciNet
    14.Ervedoza S.: Control and stabilization properties for a singular heat equation with an inverse-square potential. Commun. PDE 33, 1996–2019 (2008)CrossRef MATH MathSciNet
    15.Kogoj A., Lanconelli E.: On semilinear \({\Delta_\lambda}\) -laplace equation. Nonlinear Anal. 75, 4637–4649 (2012)CrossRef MATH MathSciNet
    16.Maz’ja V.G.: Sobolev spaces, Springer series in Soviet mathematics. Springer, Berlin (1985) (Translated from the Russian by T. O. Shaposhnikova)
    17.Morancey, M.: Approximate controllability for a 2D Grushin equation with potential having an internal singularity, arXiv:​1306.​5616 (2013).
    18.Vancostenoble J.: Carleman estimates for one-dimensional degenerate heat equations. J. Evol. Equ 6, 325–362 (2006)CrossRef MATH MathSciNet
    19.Vancostenoble J.: Improved Hardy–Poincaré inequality and shap Carleman estimates for degenerate/singular parabolic problems. Discret. Contin. Dyn. Syst. Ser. S 4, 761–190 (2011)
    20.Vancostenoble J., Zuazua E.: Null controllability of heat equations with singular inverse-square potentials. J. Funct. Anal. 254, 1864–1902 (2008)CrossRef MATH MathSciNet
  • 作者单位：Cung The Anh (1)
    Vu Manh Toi (2)

    1. Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam
    2. Faculty of Computer Science and Engineering, Hanoi Water Resources University, 175 Tay Son, Dong Da, Hanoi, Vietnam
  • 刊物类别：Mathematics and Statistics
  • 刊物主题：Mathematics
    Analysis
  • 出版者：Birkh盲user Basel
  • ISSN：1420-9004
文摘
We prove the null controllability in large time of the following linear parabolic equation involving the Grushin operator with an inverse-square potential $$u_t-\Delta_{x} u-|x|^{2}\Delta_{y}u-\frac{\mu}{|x|^2}u=v1_\omega$$in a bounded domain \({\Omega=\Omega_1\times \Omega_2\subset \mathbb{R}^{N_1} \times \mathbb{R}^{N_2} (N_1\geq 3, N_2\geq 1}\)) intersecting the surface {x = 0} under an additive control supported in an open subset \({\omega=\omega_1\times \Omega_2}\) of \({\Omega}\). Mathematics Subject Classification 93B05 93B07 35K65 35K67 Keywords Null controllability Uniform observability Grushin operator Hardy inequality Carleman inequality Dissipation speed Page %P Close Plain text Look Inside Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions Register for Journal Updates About This Journal Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Related Content Supplementary Material (0) References (20) References1.D’Ambrosio L.: Hardy inequalities related to Grushin type operators. Proc. Am. Math. Soc 132, 725–734 (2003)CrossRefMATHMathSciNet2.Alabau-Boussouira F., Cannarsa P., Fragnelli G.: Carleman estimates for weakly degenerate parabolic operators with applications to null controllability. J. Evol. Equ 6, 161–204 (2006)CrossRefMATHMathSciNet3.Anh C.T., Toi V.M.: Null controllability of a parabolic equation involving the Grushin operator in some multi-dimensional domains. Nonlinear Anal. 93, 181–196 (2013)CrossRefMATHMathSciNet4.Beauchard K., Cannarsa P., Guglielmi R.: Null controllability of Grushin-type operators in dimension two. J. Eur. Math. Soc 16, 67–101 (2014)CrossRefMATHMathSciNet5.Beauchard K., Cannarsa P., Yamamoto M.: Inverse source problem and null controllability for multidimensional parabolic operators of Grushin type. Inverse Probl. 30(2), 025006 (2014)CrossRefMATHMathSciNet6.Brezis H., Vázquez J.L.: Blowup solutions of some nonlinear elliptic problems. Rev. Mat. Univ. Complut. Madr. 10, 443–469 (1997)MATH7.Cannarsa P., Martinez P., Vancostenoble J.: Carleman estimates for a class of degenerate parabolic operators. SIAM J. Control Optim. 47, 1–19 (2008)CrossRefMATHMathSciNet8.Cannarsa P., Fragnelli G.: Null controllability of semilinear degenerate parabolic equations in bounded domains. EJDE 2006(136), 1–20 (2006)MATHMathSciNet9.Cannarsa P., Fragnelli G., Vancostenoble J.: Regional controllability of semilinear degenerate parabolic equations in bounded domains. J. Math. Anal. Appl 320, 804–818 (2006)CrossRefMATHMathSciNet10.Cannarsa P., Martinez P., Vancostenoble J.: Persistent regional controllability for a class of degenerate parabolic equations. Comm. Pure Appl. Anal. 3, 607–635 (2004)CrossRefMATHMathSciNet11.Cannarsa P., Martinez P., Vancostenoble J.: Null controllability of degenerate heat equations. Adv. Differ. Equ. 10, 153–190 (2005)MATHMathSciNet12.Cannarsa, P., Guglielmi, R.: Null controllability in large time for the parabolic Grushin operator with singular potential, geometric control theory and sub-riemannian geometry, Springer INdAM Series Volume 5, 87–102 (2014)13.Cazacu C.: Controllability of the heat equation with an inverse-square potential localized on the boundary. SIAM J. Control Optim. 52, 2055–2089 (2014)CrossRefMATHMathSciNet14.Ervedoza S.: Control and stabilization properties for a singular heat equation with an inverse-square potential. Commun. PDE 33, 1996–2019 (2008)CrossRefMATHMathSciNet15.Kogoj A., Lanconelli E.: On semilinear \({\Delta_\lambda}\) -laplace equation. Nonlinear Anal. 75, 4637–4649 (2012)CrossRefMATHMathSciNet16.Maz’ja V.G.: Sobolev spaces, Springer series in Soviet mathematics. Springer, Berlin (1985) (Translated from the Russian by T. O. Shaposhnikova)17.Morancey, M.: Approximate controllability for a 2D Grushin equation with potential having an internal singularity, arXiv:​1306.​5616 (2013).18.Vancostenoble J.: Carleman estimates for one-dimensional degenerate heat equations. J. Evol. Equ 6, 325–362 (2006)CrossRefMATHMathSciNet19.Vancostenoble J.: Improved Hardy–Poincaré inequality and shap Carleman estimates for degenerate/singular parabolic problems. Discret. Contin. Dyn. Syst. Ser. S 4, 761–190 (2011)20.Vancostenoble J., Zuazua E.: Null controllability of heat equations with singular inverse-square potentials. J. Funct. Anal. 254, 1864–1902 (2008)CrossRefMATHMathSciNet About this Article Title Null controllability in large time of a parabolic equation involving the Grushin operator with an inverse-square potential Journal Nonlinear Differential Equations and Applications NoDEA 23:20 Online DateApril 2016 DOI 10.1007/s00030-016-0364-3 Print ISSN 1021-9722 Online ISSN 1420-9004 Publisher Springer International Publishing Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Analysis Keywords 93B05 93B07 35K65 35K67 Null controllability Uniform observability Grushin operator Hardy inequality Carleman inequality Dissipation speed Authors Cung The Anh (1) Vu Manh Toi (2) Author Affiliations 1. Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam 2. Faculty of Computer Science and Engineering, Hanoi Water Resources University, 175 Tay Son, Dong Da, Hanoi, Vietnam Continue reading... 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