June  2019, 8(2): 397-422. doi: 10.3934/eect.2019020

The cost of boundary controllability for a parabolic equation with inverse square potential

Institut de Mathématiques de Toulouse, UMR CNRS 5219, Université Paul Sabatier Toulouse Ⅲ, 118 route de Narbonne, 31 062 Toulouse Cedex 4, France

Received  August 2017 Revised  November 2018 Published  March 2019

The goal of this paper is to analyze the cost of boundary null controllability for the
$ 1-D $
linear heat equation with the so-called inverse square potential:
$ u_t -u_{xx} - \frac{\mu}{x^2} u = 0, \qquad x\in (0,1), \ t \in (0,T), $
where
$ \mu $
is a real parameter such that
$ \mu \leq 1/4 $
. Since the works by Baras and Goldstein [4,5], it is known that such problems are well-posed for any
$ \mu \leq 1/4 $
(the constant appearing in the Hardy inequality) whereas instantaneous blow-up may occur when
$ \mu>1/4 $
. For any
$ \mu \leq 1/4 $
, it has been proved in [52] (via Carleman estimates) that the equation can be controlled (in any time
$ T>0 $
) by a locally distributed control. Obviously, the same result holds true when one considers the case of a boundary control acting at
$ x = 1 $
. The goal of the present paper is to provide sharp estimates of the cost of the control in that case, analyzing its dependence with respect to the two paramaters
$ T>0 $
and
$ \mu \in (-\infty, 1/4] $
. Our proofs are based on the moment method and very recent results on biorthogonal sequences.
Citation: Patrick Martinez, Judith Vancostenoble. The cost of boundary controllability for a parabolic equation with inverse square potential. Evolution Equations & Control Theory, 2019, 8 (2) : 397-422. doi: 10.3934/eect.2019020
References:
[1]

F. Ammar KhodjaA. BenabdallahM. González-Burgos and L. de Teresa, The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials, J. Math. Pures Appl., 96 (2011), 555-590.  doi: 10.1016/j.matpur.2011.06.005.  Google Scholar

[2]

G. Avalos and I. Lasiecka, Optimal blowup rates for the minimal energy null control of the strongly damped abstract wave equation, Ann. Sc. Norm. Super. Pisa Cl. Sci., 2 (2003), 601-616.   Google Scholar

[3]

G. Avalos and I. Lasiecka, The null controllability of thermoelastic plates and singularity of the associated minimal energy function, J. Math. Anal. Appl., 294 (2004), 34-61.  doi: 10.1016/j.jmaa.2004.01.035.  Google Scholar

[4]

P. Baras and J. Goldstein, Remarks on the inverse square potential in quantum mechanics, Differential Equations, 92 (1984), 31-35.  doi: 10.1016/S0304-0208(08)73675-2.  Google Scholar

[5]

P. Baras and J. Goldstein, The heat equation with a singular potential, Trans. Amer. Math. Soc., 284 (1984), 121-139.  doi: 10.1090/S0002-9947-1984-0742415-3.  Google Scholar

[6]

A. BenabdallahF. BoyerM. González-Burgos and G. Olive, Sharp estimates of the one-dimensional boundary control cost for parabolic systems and applications to the $N$-dimensional boundary null controllability in cylindrical domains, SIAM J. Control Optim., 52 (2014), 2970-3001.  doi: 10.1137/130929680.  Google Scholar

[7]

U. Biccari, V. Hernández-Santamaría and J. Vancostenoble, Existence and cost of boundary controls for a degenerate/singular parabolic equation, preprint. Google Scholar

[8]

U. Biccari and E. Zuazua, Null controllability for a heat equation with a singular inverse-square potential involving the distance to the boundary function, J. Differential Equations, 261 (2016), 2809-2853.  doi: 10.1016/j.jde.2016.05.019.  Google Scholar

[9]

P. CannarsaP. Martinez and J. Vancostenoble, The cost of controlling weakly degenerate parabolic equations by boundary controls, Math. Control Relat. Fields, 7 (2017), 171-211.  doi: 10.3934/mcrf.2017006.  Google Scholar

[10]

P. Cannarsa, P. Martinez and J. Vancostenoble, Precise estimates for biorthogonal families under asymptotic gap conditions, DCDS-s, in press. Google Scholar

[11]

P. Cannarsa, P. Martinez and J. Vancostenoble, The cost of controlling strongly degenerate parabolic equations, ESAIM: Control, Optimisation and Calculus of Variations, in press, 2018. doi: 10.1051/cocv/2018007.  Google Scholar

[12]

C. Cazacu, Controllability of the heat equation with an inverse-square potential localized on the boundary, SIAM Journal of Control and Optimization, 52 (2014), 2055-2089.  doi: 10.1137/120862557.  Google Scholar

[13]

J. M. Coron and S. Guerrero, Singular optimal control: A linear 1-D parabolic-hyperbolic example, Asymp. Anal., 44 (2005), 237-257.   Google Scholar

[14]

G. Da Prato, An introduction to Infinite Dimensional Analysis, Scuola Normale Superiore, 2001.  Google Scholar

[15] G. Da Prato and J. Zabczyk, Second Order Partial Differential Equations in Hilbert Spaces, London Mathematical Society Lecture Note Series, 293. Cambridge University Press, Cambridge, 2002.  doi: 10.1017/CBO9780511543210.  Google Scholar
[16]

S. Ervedoza, Control and stabilization properties for a singular heat equation with an inverse-square potential, Comm. Partial Differential Equations, 33 (2008), 1996-2019.  doi: 10.1080/03605300802402633.  Google Scholar

[17]

W. N. Everitt, A catalogue of Sturm-Liouville differential equations, Sturm-Liouville Theory, Birkhäuser, Basel, (2005), 271–331. doi: 10.1007/3-7643-7359-8_12.  Google Scholar

[18]

H. O. Fattorini and D. L. Russel, Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Rat. Mech. Anal., 43 (1971), 272-292.  doi: 10.1007/BF00250466.  Google Scholar

[19]

H. O. Fattorini and D. L. Russel, Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations, Quart. Appl. Math., 32 (1974/75), 45-69.  doi: 10.1090/qam/510972.  Google Scholar

[20]

E. Fernandez-Cara and E. Zuazua, The cost of approximate controllability for heat equations: The linear case, Adv. Differential equations, 5 (2000), 465-514.   Google Scholar

[21]

M. Fotouhi and L. Salimi, Null controllability of degenerate/singular parabolic equations, Journal of Dynamical and Control Systems, 18 (2012), 573-602.  doi: 10.1007/s10883-012-9160-5.  Google Scholar

[22]

A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Ser. 34, Seoul National University, Seoul, Korea, 1996.  Google Scholar

[23]

O. Glass, A complex-analytic approach to the problem of uniform controllability of transport equation in the vanishing viscosity limit, J. Funct. Anal., 258 (2010), 852-868.  doi: 10.1016/j.jfa.2009.06.035.  Google Scholar

[24]

F. Gozzi and P. Loreti, Regularity of the minimum time function and minimum energy problem, SIAM J. Control Optim., 37 (1999), 1195-1221.  doi: 10.1137/S0363012996312763.  Google Scholar

[25]

M. Gueye, Exact boundary controllability of 1-D parabolic and hyperbolic degenerate equations, SIAM J. Control Optim., 52 (2014), 2037-2054.  doi: 10.1137/120901374.  Google Scholar

[26]

E. N. Güichal, A lower bound of the norm of the control operator for the heat equation, Journal of Mathematical Analysis and Applications, 110 (1985), 519-527.  doi: 10.1016/0022-247X(85)90313-0.  Google Scholar

[27]

A. HajjajL. Maniar and J. Salhi, Carleman estimates and null controllability of degenerate/singular parabolic systems, Electronic Journal of Di erential Equations, 292 (2016), 1-25.   Google Scholar

[28]

S. Hansen, Bounds on functions biorthogonal to sets of complex exponentials; control of damped elastic systems, Journal of Math. Anal. and Appl., 158 (1991), 487-508.  doi: 10.1016/0022-247X(91)90252-U.  Google Scholar

[29] G. H. HardyJ. E. Littlewood and G. Pólya, Inequalities, 2d ed. Cambridge, at the University Press, 1952.   Google Scholar
[30]

E. Kamke, Differentialgleichungen: Lösungsmethoden und Lösungen. Band 1: Gewöhnliche Differentialgleichungen, Neunte Auflage. Mit einem Vorwort von Detlef Kamke. B. G. Teubner, Stuttgart, 1977.  Google Scholar

[31]

V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer, Berlin, 2005.  Google Scholar

[32]

I. Lasiecka and T. I. Seidman, Blowup estimates for observability of a thermoelastic system, Asymptot. Anal., 50 (2006), 93-120.   Google Scholar

[33]

G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. Partial Differential Equations, 20 (1995), 335-356.  doi: 10.1080/03605309508821097.  Google Scholar

[34]

N. N. Lebedev, Special Functions and their Applications, Dover Publications, New York, 1972.  Google Scholar

[35]

P. Lissy, On the cost of fast controls for some families of dispersive or parabolic equations in one space dimension, SIAM J. Control Optim., 52 (2014), 2651-2676.  doi: 10.1137/140951746.  Google Scholar

[36]

P. Lissy, Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport-diffusion equation, J. Differential Equations, 259 (2015), 5331-5352.  doi: 10.1016/j.jde.2015.06.031.  Google Scholar

[37]

L. Lorch and M. E. Muldoon, Monotonic sequences related to zeros of Bessel functions, Numer. Algor., 49 (2008), 221-233.  doi: 10.1007/s11075-008-9189-4.  Google Scholar

[38]

P. MartinL. Rosier and P. Rouchon, Null controllability of one-dimensional parabolic equations by the flatness approach, SIAM J. Control Optim., 54 (2016), 198-220.  doi: 10.1137/14099245X.  Google Scholar

[39]

L. Miller, Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time, J. Differential Equations, 204 (2004), 202-226.  doi: 10.1016/j.jde.2004.05.007.  Google Scholar

[40]

L. Miller, How violent are fast controls for Schrödinger and plate vibrations?, Arch. Ration. Mech. Anal., 172 (2004), 429-456.  doi: 10.1007/s00205-004-0312-y.  Google Scholar

[41]

L. Miller, The control transmutation method and the cost of fast controls, SIAM J. Control Optim., 45 (2006), 762-772.  doi: 10.1137/S0363012904440654.  Google Scholar

[42]

B. Opic and A. Kufner, Hardy-type Inequalities, Pitman Research Notes in Math., Vol. 219, Longman, 1990.  Google Scholar

[43]

Y. PrivatE. Trélat and E. Zuazua, Optimal observation of the one-dimensional wave equation, J. Fourier Anal. Appl., 19 (2013), 514-544.  doi: 10.1007/s00041-013-9267-4.  Google Scholar

[44]

C. K. Qu and R. Wong, "Best possible'' upper and lower bounds for the zeros of the Bessel function $J_\nu(x)$, Trans. Amer. Math. Soc., 351 (1999), 2833-2859.  doi: 10.1090/S0002-9947-99-02165-0.  Google Scholar

[45]

T. I. Seidman, Two results on exact boundary control of parabolic equations, Appl. Math. Optim., 11 (1984), 145-152.  doi: 10.1007/BF01442174.  Google Scholar

[46]

T. I. Seidman, How violent are fast controls?, Math. Control Signals Systems, 1 (1988), 89-95.  doi: 10.1007/BF02551238.  Google Scholar

[47]

T. I. Seidman and J. Yong, How violent are fast controls? Ⅱ, Math. Control Signals Systems, 9 (1996), 327-340.  doi: 10.1007/BF01211854.  Google Scholar

[48]

T. I. SeidmanS. A. Avdonin and S. A. Ivanov, The "window problem" for series of complex exponentials, J. Fourier Anal. Appl., 6 (2000), 233-254.  doi: 10.1007/BF02511154.  Google Scholar

[49]

G. Tenenbaum and M. Tucsnak, New blow-up rates for fast controls of Schrodinger and heat equations, J. Differential Equations, 243 (2007), 70-100.  doi: 10.1016/j.jde.2007.06.019.  Google Scholar

[50]

J. Vancostenoble, Improved Hardy-Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems, Discrete and Continuous Dynamical Systems - Series S, American Institute of Mathematical Sciences, 4 (2011), 761-790.  doi: 10.3934/dcdss.2011.4.761.  Google Scholar

[51]

J. Vancostenoble, Lipschitz stability in inverse source problems for singular parabolic equations, Communications in Partial Differential Equations, 36 (2011), 1287-1317.  doi: 10.1080/03605302.2011.587491.  Google Scholar

[52]

J. Vancostenoble and E. Zuazua, Null controllability for the heat equation with singular inverse-square potentials, J. Funct. Anal., 254 (2008), 1864-1902.  doi: 10.1016/j.jfa.2007.12.015.  Google Scholar

[53]

J. L. Vázquez and E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal., 173 (2000), 103-153.  doi: 10.1006/jfan.1999.3556.  Google Scholar

[54] G. N. Watson, A Treatise on the Theory of Bessel Functions, second edition, Cambridge University Press, Cambridge, England, 1944.   Google Scholar

show all references

References:
[1]

F. Ammar KhodjaA. BenabdallahM. González-Burgos and L. de Teresa, The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials, J. Math. Pures Appl., 96 (2011), 555-590.  doi: 10.1016/j.matpur.2011.06.005.  Google Scholar

[2]

G. Avalos and I. Lasiecka, Optimal blowup rates for the minimal energy null control of the strongly damped abstract wave equation, Ann. Sc. Norm. Super. Pisa Cl. Sci., 2 (2003), 601-616.   Google Scholar

[3]

G. Avalos and I. Lasiecka, The null controllability of thermoelastic plates and singularity of the associated minimal energy function, J. Math. Anal. Appl., 294 (2004), 34-61.  doi: 10.1016/j.jmaa.2004.01.035.  Google Scholar

[4]

P. Baras and J. Goldstein, Remarks on the inverse square potential in quantum mechanics, Differential Equations, 92 (1984), 31-35.  doi: 10.1016/S0304-0208(08)73675-2.  Google Scholar

[5]

P. Baras and J. Goldstein, The heat equation with a singular potential, Trans. Amer. Math. Soc., 284 (1984), 121-139.  doi: 10.1090/S0002-9947-1984-0742415-3.  Google Scholar

[6]

A. BenabdallahF. BoyerM. González-Burgos and G. Olive, Sharp estimates of the one-dimensional boundary control cost for parabolic systems and applications to the $N$-dimensional boundary null controllability in cylindrical domains, SIAM J. Control Optim., 52 (2014), 2970-3001.  doi: 10.1137/130929680.  Google Scholar

[7]

U. Biccari, V. Hernández-Santamaría and J. Vancostenoble, Existence and cost of boundary controls for a degenerate/singular parabolic equation, preprint. Google Scholar

[8]

U. Biccari and E. Zuazua, Null controllability for a heat equation with a singular inverse-square potential involving the distance to the boundary function, J. Differential Equations, 261 (2016), 2809-2853.  doi: 10.1016/j.jde.2016.05.019.  Google Scholar

[9]

P. CannarsaP. Martinez and J. Vancostenoble, The cost of controlling weakly degenerate parabolic equations by boundary controls, Math. Control Relat. Fields, 7 (2017), 171-211.  doi: 10.3934/mcrf.2017006.  Google Scholar

[10]

P. Cannarsa, P. Martinez and J. Vancostenoble, Precise estimates for biorthogonal families under asymptotic gap conditions, DCDS-s, in press. Google Scholar

[11]

P. Cannarsa, P. Martinez and J. Vancostenoble, The cost of controlling strongly degenerate parabolic equations, ESAIM: Control, Optimisation and Calculus of Variations, in press, 2018. doi: 10.1051/cocv/2018007.  Google Scholar

[12]

C. Cazacu, Controllability of the heat equation with an inverse-square potential localized on the boundary, SIAM Journal of Control and Optimization, 52 (2014), 2055-2089.  doi: 10.1137/120862557.  Google Scholar

[13]

J. M. Coron and S. Guerrero, Singular optimal control: A linear 1-D parabolic-hyperbolic example, Asymp. Anal., 44 (2005), 237-257.   Google Scholar

[14]

G. Da Prato, An introduction to Infinite Dimensional Analysis, Scuola Normale Superiore, 2001.  Google Scholar

[15] G. Da Prato and J. Zabczyk, Second Order Partial Differential Equations in Hilbert Spaces, London Mathematical Society Lecture Note Series, 293. Cambridge University Press, Cambridge, 2002.  doi: 10.1017/CBO9780511543210.  Google Scholar
[16]

S. Ervedoza, Control and stabilization properties for a singular heat equation with an inverse-square potential, Comm. Partial Differential Equations, 33 (2008), 1996-2019.  doi: 10.1080/03605300802402633.  Google Scholar

[17]

W. N. Everitt, A catalogue of Sturm-Liouville differential equations, Sturm-Liouville Theory, Birkhäuser, Basel, (2005), 271–331. doi: 10.1007/3-7643-7359-8_12.  Google Scholar

[18]

H. O. Fattorini and D. L. Russel, Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Rat. Mech. Anal., 43 (1971), 272-292.  doi: 10.1007/BF00250466.  Google Scholar

[19]

H. O. Fattorini and D. L. Russel, Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations, Quart. Appl. Math., 32 (1974/75), 45-69.  doi: 10.1090/qam/510972.  Google Scholar

[20]

E. Fernandez-Cara and E. Zuazua, The cost of approximate controllability for heat equations: The linear case, Adv. Differential equations, 5 (2000), 465-514.   Google Scholar

[21]

M. Fotouhi and L. Salimi, Null controllability of degenerate/singular parabolic equations, Journal of Dynamical and Control Systems, 18 (2012), 573-602.  doi: 10.1007/s10883-012-9160-5.  Google Scholar

[22]

A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Ser. 34, Seoul National University, Seoul, Korea, 1996.  Google Scholar

[23]

O. Glass, A complex-analytic approach to the problem of uniform controllability of transport equation in the vanishing viscosity limit, J. Funct. Anal., 258 (2010), 852-868.  doi: 10.1016/j.jfa.2009.06.035.  Google Scholar

[24]

F. Gozzi and P. Loreti, Regularity of the minimum time function and minimum energy problem, SIAM J. Control Optim., 37 (1999), 1195-1221.  doi: 10.1137/S0363012996312763.  Google Scholar

[25]

M. Gueye, Exact boundary controllability of 1-D parabolic and hyperbolic degenerate equations, SIAM J. Control Optim., 52 (2014), 2037-2054.  doi: 10.1137/120901374.  Google Scholar

[26]

E. N. Güichal, A lower bound of the norm of the control operator for the heat equation, Journal of Mathematical Analysis and Applications, 110 (1985), 519-527.  doi: 10.1016/0022-247X(85)90313-0.  Google Scholar

[27]

A. HajjajL. Maniar and J. Salhi, Carleman estimates and null controllability of degenerate/singular parabolic systems, Electronic Journal of Di erential Equations, 292 (2016), 1-25.   Google Scholar

[28]

S. Hansen, Bounds on functions biorthogonal to sets of complex exponentials; control of damped elastic systems, Journal of Math. Anal. and Appl., 158 (1991), 487-508.  doi: 10.1016/0022-247X(91)90252-U.  Google Scholar

[29] G. H. HardyJ. E. Littlewood and G. Pólya, Inequalities, 2d ed. Cambridge, at the University Press, 1952.   Google Scholar
[30]

E. Kamke, Differentialgleichungen: Lösungsmethoden und Lösungen. Band 1: Gewöhnliche Differentialgleichungen, Neunte Auflage. Mit einem Vorwort von Detlef Kamke. B. G. Teubner, Stuttgart, 1977.  Google Scholar

[31]

V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer, Berlin, 2005.  Google Scholar

[32]

I. Lasiecka and T. I. Seidman, Blowup estimates for observability of a thermoelastic system, Asymptot. Anal., 50 (2006), 93-120.   Google Scholar

[33]

G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. Partial Differential Equations, 20 (1995), 335-356.  doi: 10.1080/03605309508821097.  Google Scholar

[34]

N. N. Lebedev, Special Functions and their Applications, Dover Publications, New York, 1972.  Google Scholar

[35]

P. Lissy, On the cost of fast controls for some families of dispersive or parabolic equations in one space dimension, SIAM J. Control Optim., 52 (2014), 2651-2676.  doi: 10.1137/140951746.  Google Scholar

[36]

P. Lissy, Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport-diffusion equation, J. Differential Equations, 259 (2015), 5331-5352.  doi: 10.1016/j.jde.2015.06.031.  Google Scholar

[37]

L. Lorch and M. E. Muldoon, Monotonic sequences related to zeros of Bessel functions, Numer. Algor., 49 (2008), 221-233.  doi: 10.1007/s11075-008-9189-4.  Google Scholar

[38]

P. MartinL. Rosier and P. Rouchon, Null controllability of one-dimensional parabolic equations by the flatness approach, SIAM J. Control Optim., 54 (2016), 198-220.  doi: 10.1137/14099245X.  Google Scholar

[39]

L. Miller, Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time, J. Differential Equations, 204 (2004), 202-226.  doi: 10.1016/j.jde.2004.05.007.  Google Scholar

[40]

L. Miller, How violent are fast controls for Schrödinger and plate vibrations?, Arch. Ration. Mech. Anal., 172 (2004), 429-456.  doi: 10.1007/s00205-004-0312-y.  Google Scholar

[41]

L. Miller, The control transmutation method and the cost of fast controls, SIAM J. Control Optim., 45 (2006), 762-772.  doi: 10.1137/S0363012904440654.  Google Scholar

[42]

B. Opic and A. Kufner, Hardy-type Inequalities, Pitman Research Notes in Math., Vol. 219, Longman, 1990.  Google Scholar

[43]

Y. PrivatE. Trélat and E. Zuazua, Optimal observation of the one-dimensional wave equation, J. Fourier Anal. Appl., 19 (2013), 514-544.  doi: 10.1007/s00041-013-9267-4.  Google Scholar

[44]

C. K. Qu and R. Wong, "Best possible'' upper and lower bounds for the zeros of the Bessel function $J_\nu(x)$, Trans. Amer. Math. Soc., 351 (1999), 2833-2859.  doi: 10.1090/S0002-9947-99-02165-0.  Google Scholar

[45]

T. I. Seidman, Two results on exact boundary control of parabolic equations, Appl. Math. Optim., 11 (1984), 145-152.  doi: 10.1007/BF01442174.  Google Scholar

[46]

T. I. Seidman, How violent are fast controls?, Math. Control Signals Systems, 1 (1988), 89-95.  doi: 10.1007/BF02551238.  Google Scholar

[47]

T. I. Seidman and J. Yong, How violent are fast controls? Ⅱ, Math. Control Signals Systems, 9 (1996), 327-340.  doi: 10.1007/BF01211854.  Google Scholar

[48]

T. I. SeidmanS. A. Avdonin and S. A. Ivanov, The "window problem" for series of complex exponentials, J. Fourier Anal. Appl., 6 (2000), 233-254.  doi: 10.1007/BF02511154.  Google Scholar

[49]

G. Tenenbaum and M. Tucsnak, New blow-up rates for fast controls of Schrodinger and heat equations, J. Differential Equations, 243 (2007), 70-100.  doi: 10.1016/j.jde.2007.06.019.  Google Scholar

[50]

J. Vancostenoble, Improved Hardy-Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems, Discrete and Continuous Dynamical Systems - Series S, American Institute of Mathematical Sciences, 4 (2011), 761-790.  doi: 10.3934/dcdss.2011.4.761.  Google Scholar

[51]

J. Vancostenoble, Lipschitz stability in inverse source problems for singular parabolic equations, Communications in Partial Differential Equations, 36 (2011), 1287-1317.  doi: 10.1080/03605302.2011.587491.  Google Scholar

[52]

J. Vancostenoble and E. Zuazua, Null controllability for the heat equation with singular inverse-square potentials, J. Funct. Anal., 254 (2008), 1864-1902.  doi: 10.1016/j.jfa.2007.12.015.  Google Scholar

[53]

J. L. Vázquez and E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal., 173 (2000), 103-153.  doi: 10.1006/jfan.1999.3556.  Google Scholar

[54] G. N. Watson, A Treatise on the Theory of Bessel Functions, second edition, Cambridge University Press, Cambridge, England, 1944.   Google Scholar
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