Opuscula Math. 39, no. 2 (2019), 207-225
https://doi.org/10.7494/OpMath.2019.39.2.207

 
Opuscula Mathematica

Controllability of degenerate and singular parabolic problems: the double strong case with Neumann boundary conditions

Genni Fragnelli
Dimitri Mugnai

Abstract. We prove a null controllability result for a parabolic problem with Neumann boundary conditions. We consider non smooth coefficients in presence of a strongly singular potential and a strongly degenerate coefficient, both vanishing at an interior point. This paper concludes the study of the Neumann case.

Keywords: strongly singular/degenerate equations, non smooth coefficients, null controllability.

Mathematics Subject Classification: 35Q93, 93B05, 34H05.

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  1. F. Alabau-Boussouira, P. Cannarsa, G Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability, J. Evol. Eqs 6 (2006), 161-204.
  2. K. Beauchard, P. Cannarsa, R. Guglielmi, Null controllability of Grushin-type operators in dimension two, J. Eur. Math. Soc. (JEMS) 16 (2014), 67-101.
  3. I. Boutaayamou, G. Fragnelli, L. Maniar, Lipschitz stability for linear cascade parabolic systems with interior degeneracy, Electron. J. Diff. Equ. 2014 (2014), 1-26.
  4. I. Boutaayamou, G. Fragnelli, L. Maniar, Carleman estimates for parabolic equations with interior degeneracy and Neumann boundary conditions, J. Anal. Math. 135 (2018), 1-35.
  5. P. Cannarsa, G. Fragnelli, D. Rocchetti, Null controllability of degenerate parabolic operators with drift, Netw. Heterog. Media 2 (2007), 693-713.
  6. P. Cannarsa, G. Fragnelli, D. Rocchetti, Controllability results for a class of one-dimensional degenerate parabolic problems in nondivergence form, J. Evol. Equ. 8 (2008), 583-616.
  7. P. Cannarsa, P. Martinez, J. Vancostenoble, Null controllability of the degenerate heat equations, Adv. Differential Equations 10 (2005), 153-190.
  8. P. Cannarsa, P. Martinez, J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators, SIAM J. Control Optim. 47 (2008), 1-19.
  9. S. Ervedoza, Null Controllability for a singular heat equation: Carleman estimates and Hardy inequalities, Comm. Partial Differential Equations 33 (2008), 1996-2019.
  10. G. Floridia, Approximate controllability for nonlinear degenerate parabolic problems with bilinear control, J. Differential Equations 257 (2014), 3382-3422.
  11. M. Fotouhi, L. Salimi, Null controllability of degenerate/singular parabolic equations, J. Dyn. Control Syst. 18 (2012), 573-602.
  12. M. Fotouhi, L. Salimi, Controllability results for a class of one dimensional degenerate/singular parabolic equations, Commun. Pure Appl. Anal. 12 (2013), 1415-1430.
  13. G. Fragnelli, Null controllability of degenerate parabolic equations in non divergence form via Carleman estimates, Discrete Contin. Dyn. Syst. Ser. S 6 (2013), 687-701.
  14. G. Fragnelli, Interior degenerate/singular parabolic equations in nondivergence form: well-posedness and Carleman estimates, J. Differential Equations 260 (2016), 1314-1371.
  15. G. Fragnelli, D. Mugnai, Carleman estimates and observability inequalities for parabolic equations with interior degeneracy, Advances in Nonlinear Analysis 2 (2013), 339-378.
  16. G. Fragnelli, D. Mugnai, Carleman estimates, observability inequalities and null controllability for interior degenerate non smooth parabolic equations, Mem. Amer. Math. Soc. 242 (2016) 1146.
  17. G. Fragnelli, D. Mugnai, Corrigendum and improvements to "Carleman estimates, observability inequalities and null controllability for interior degenerate non smooth parabolic equations, and its consequences", to appear in Mem. Amer. Math. Soc.
  18. G. Fragnelli, D. Mugnai, Carleman estimates for singular parabolic equations with interior degeneracy and non smooth coefficients, Adv. Nonlinear Anal. 6 (2017), 61-84.
  19. G. Fragnelli, D. Mugnai, Singular parabolic equations with interior degeneracy and non smooth coefficients: the Neumann case, to appear in Discrete Contin. Dyn. Syst. Ser. S.
  20. G. Fragnelli, D. Mugnai, Controllability of strongly degenerate parabolic problems with strongly singular potentials, Electron. J. Qual. Theory Differ. Equ. 2018, no. 50 , 1-11.
  21. G. Fragnelli, G. Ruiz Goldstein, J.A. Goldstein, S. Romanelli, Generators with interior degeneracy on spaces of \(L^2\) type, Electron. J. Differential Equations 2012 (2012), 1-30.
  22. A. Hajjaj, L. Maniar, J. Salhi, Carleman estimates and null controllability of degenerate/singular parabolic systems, Electron. J. Differential Equations 2016 (2016) 292, pp. 1-25.
  23. H. Koch, D. Tataru, Carleman estimates and unique continuation for second order parabolic equations with nonsmooth coefficients, Comm. Partial Differential Equations 34 (2009), 305-366.
  24. S. Micu, E. Zuazua, On the lack of null controllability of the heat equation on the half-line, Trans. Amer. Math. Soc. 353 (2001), 1635-1659.
  25. B. Muckenhoupt, Hardy's inequality with weights, Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, I. Studia Math. 44 (1972), 31-38.
  26. M. Renardy, R.C. Rogers, An Introduction to Partial Differential Equations, 2nd ed., Texts Appl. Math. 13, Springer, New York, 2004.
  27. D.D. Repovš, The Ambrosetti-Prodi problem with degenerate potential and Neumann boundary condition, arXiv:1802.03194.
  28. J. Vancostenoble, Improved Hardy-Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems, Discrete Contin. Dyn. Syst. Ser. S 4 (2011), 761-790.
  29. J. Vancostenoble, E. Zuazua, Null controllability for the heat equation with singular inverse-square potentials, J. Funct. Anal. 254 (2008), 1864-1902.
  • Communicated by Giovanni Molica Bisci.
  • Received: 2018-06-25.
  • Accepted: 2018-11-03.
  • Published online: 2018-12-07.
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Cite this article as:
Genni Fragnelli, Dimitri Mugnai, Controllability of degenerate and singular parabolic problems: the double strong case with Neumann boundary conditions, Opuscula Math. 39, no. 2 (2019), 207-225, https://doi.org/10.7494/OpMath.2019.39.2.207

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