December  2019, 9(4): 673-696. doi: 10.3934/mcrf.2019046

Backward uniqueness results for some parabolic equations in an infinite rod

Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, CNRS, UPS IMT F-31062 Toulouse Cedex 9, France

* Corresponding author: Sylvain Ervedoza

Received  June 2018 Revised  August 2019 Published  November 2019

Fund Project: The first author is partially supported by IFSMACS ANR-15-CE40-0010 of the French National Research Agency (ANR) and both authors are supported by the CIMI Labex, Toulouse, France, under grant ANR-11-LABX-0040-CIMI.

The goal of this article is to provide backward uniqueness results for several models of parabolic equations set on the half line, namely the heat equation, and the heat equation with quadratic potential and with purely imaginary quadratic potentials, with non-homogeneous boundary conditions. Such result can thus also be interpreted as a strong lack of controllability on the half line, as it shows that only the trivial initial datum can be steered to zero. Our results are based on the explicit knowledge of the kernel of each equation, and standard arguments from complex analysis, namely the Phragmén-Lindelöf principle.

Citation: Jérémi Dardé, Sylvain Ervedoza. Backward uniqueness results for some parabolic equations in an infinite rod. Mathematical Control and Related Fields, 2019, 9 (4) : 673-696. doi: 10.3934/mcrf.2019046
References:
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H. AikawaN. Hayashi and S. Saitoh, The Bergman space on a sector and the heat equation, Complex Variables Theory Appl., 15 (1990), 27-36.  doi: 10.1080/17476939008814430.

[2]

K. Beauchard, Null controllability of Kolmogorov-type equations, Math. Control Signals Systems, 26 (2014), 145-176.  doi: 10.1007/s00498-013-0110-x.

[3]

K. BeauchardP. Cannarsa and R. Guglielmi, Null controllability of Grushin-type operators in dimension two, J. Eur. Math. Soc. (JEMS), 16 (2014), 67-101.  doi: 10.4171/JEMS/428.

[4]

K. Beauchard, J. Dardé and S. Ervedoza, Minimal time issues for the observability of Grushin-type equations, December 2017.

[5]

K. BeauchardB. HelfferR. Henry and L. Robbiano, Degenerate parabolic operators of Kolmogorov type with a geometric control condition, ESAIM Control Optim. Calc. Var., 21 (2015), 487-512.  doi: 10.1051/cocv/2014035.

[6]

K. BeauchardL. Miller and M. Morancey, 2D Grushin-type equations: Minimal time and null controllable data, J. Differential Equations, 259 (2015), 5813-5845.  doi: 10.1016/j.jde.2015.07.007.

[7]

K. Beauchard and K. Pravda-Starov, Null-controllability of non-autonomous Ornstein-Uhlenbeck equations, J. Math. Anal. Appl., 456 (2017), 496–524. doi: 10.1016/j.jmaa.2017.07.014.

[8]

L. S. Boulton, Non-self-adjoint harmonic oscillator, compact semigroups and pseudospectra, J. Operator Theory, 47 (2002), 413-429. 

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[10]

E. B. Davies, Pseudo-spectra, the harmonic oscillator and complex resonances, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 455 (1999), 585-599.  doi: 10.1098/rspa.1999.0325.

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E. B. Davies, Semi-classical states for non-self-adjoint Schrödinger operators, Comm. Math. Phys., 200 (1999), 35-41.  doi: 10.1007/s002200050521.

[12]

E. B. Davies and A. B. J. Kuijlaars, Spectral asymptotics of the non-self-adjoint harmonic oscillator, J. London Math. Soc. (2), 70 (2004), 420-426.  doi: 10.1112/S0024610704005381.

[13]

T. Duyckaerts and L. Miller, Resolvent conditions for the control of parabolic equations, J. Funct. Anal., 263 (2012), 3641-3673.  doi: 10.1016/j.jfa.2012.09.003.

[14]

Ju. V. Egorov, Some problems in the theory of optimal control, Ž. Vyčisl. Mat. i Mat. Fiz., 3 (1963), 887-904. 

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L. EscauriazaC. E. KenigG. Ponce and L. Vega, Hardy's uncertainty principle, convexity and Schrödinger evolutions, J. Eur. Math. Soc. (JEMS), 10 (2008), 883-907.  doi: 10.4171/JEMS/134.

[16]

L. EscauriazaG. Seregin and V. Šverák, Backward uniqueness for the heat operator in half-space, Algebra i Analiz, 15 (2003), 201-214. 

[17]

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

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

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A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, volume 34 of Lecture Notes Series, Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul, 1996.

[20]

B. Helffer, Semi-classical Analysis for the Schrödinger Operator and Applications, volume 1336 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1988. doi: 10.1007/BFb0078115.

[21]

L. IskauriazaG. A. Serëgin and V. Shverak, $L_{3, \infty}$-solutions of Navier-Stokes equations and backward uniqueness, Uspekhi Mat. Nauk, 58 (2003), 3-44.  doi: 10.1070/RM2003v058n02ABEH000609.

[22]

F. John, Partial Differential Equations, volume 1 of Applied Mathematical Sciences, Springer-Verlag, New York, fourth edition, 1982. doi: 10.1007/978-1-4684-9333-7.

[23]

B. Frank Jones and Jr ., A fundamental solution for the heat equation which is supported in a strip, J. Math. Anal. Appl., 60 (1977), 314-324.  doi: 10.1016/0022-247X(77)90021-X.

[24]

A. Koenig, Non null controllability of the Grushin equation in 2D, C. R. Math. Acad. Sci. Paris, 355 (2017), 1215–1235. https://arXiv.org/abs/1701.06467, 2017. doi: 10.1016/j.crma.2017.10.021.

[25]

C. Laurent and M. Leautaud, Tunneling estimates and approximate controllability for hypoelliptic equations, https://arXiv.org/abs/1703.10797, March 2017.

[26]

J. Le Rousseau and I. Moyano, Null-controllability of the Kolmogorov equation in the whole phase space, J. Differential Equations, 260 (2016), 3193-3233.  doi: 10.1016/j.jde.2015.09.062.

[27]

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.

[28]

B. Ya. Levin, Lectures on Entire Functions, volume 150 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1996. In collaboration with and with a preface by Yu. Lyubarskii, M. Sodin and V. Tkachenko, Translated from the Russian manuscript by Tkachenko.

[29]

L. Li and V. Šverák, Backward uniqueness for the heat equation in cones, Comm. Partial Differential Equations, 37 (2012), 1414-1429.  doi: 10.1080/03605302.2011.635323.

[30]

S. Micu and E. Zuazua, On the lack of null-controllability of the heat equation on the half-line, Trans. Amer. Math. Soc., 353 (2001), 1635–1659 (electronic). doi: 10.1090/S0002-9947-00-02665-9.

[31]

S. Micu and E. Zuazua, On the controllability of a fractional order parabolic equation, SIAM J. Control Optim., 44 (2006), 1950–1972 (electronic). doi: 10.1137/S036301290444263X.

[32]

L. Miller, On the null-controllability of the heat equation in unbounded domains, Bull. Sci. Math., 129 (2005), 175-185.  doi: 10.1016/j.bulsci.2004.04.003.

[33]

L. Miller, Unique continuation estimates for the Laplacian and the heat equation on non-compact manifolds, Math. Res. Lett., 12 (2005), 37-47.  doi: 10.4310/MRL.2005.v12.n1.a4.

[34]

L. Miller, On the controllability of anomalous diffusions generated by the fractional Laplacian, Math. Control Signals Systems, 18 (2006), 260-271.  doi: 10.1007/s00498-006-0003-3.

[35]

L. Miller, Unique continuation estimates for sums of semiclassical eigenfunctions and null-controllability from cones, 20 pages, 1 figure, AMS-LaTeX., November 2008.

[36]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, volume 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[37]

J. Rauch, Partial Differential Equations, volume 128 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0953-9.

[38]

A. Rüland, On the backward uniqueness property for the heat equation in two-dimensional conical domains, Manuscripta Math., 147 (2015), 415-436.  doi: 10.1007/s00229-015-0764-4.

[39]

G. Seregin and V. Šverák, The Navier-Stokes equations and backward uniqueness, In Nonlinear Problems in Mathematical Physics and Related Topics, II, volume 2 of Int. Math. Ser. (N. Y.), pages 353–366. Kluwer/Plenum, New York, 2002.

[40]

G. Wang, M. Wang, C. Zhang and Y. Zhang, Observable set, observability, interpolation inequality and spectral inequality for the heat equation in $\mathbb{R}^n$, J. Math. Pures Appl., 126 (2019), 144–194, https://arXiv.org/abs/1711.04279. doi: 10.1016/j.matpur.2019.04.009.

[41]

D. V. Widder, The role of the Appell transformation in the theory of heat conduction, Trans. Amer. Math. Soc., 109 (1963), 121-134.  doi: 10.1090/S0002-9947-1963-0154068-2.

[42]

J. Wu and W. Wang, On backward uniqueness for the heat operator in cones, J. Differential Equations, 258 (2015), 224-241.  doi: 10.1016/j.jde.2014.09.011.

show all references

References:
[1]

H. AikawaN. Hayashi and S. Saitoh, The Bergman space on a sector and the heat equation, Complex Variables Theory Appl., 15 (1990), 27-36.  doi: 10.1080/17476939008814430.

[2]

K. Beauchard, Null controllability of Kolmogorov-type equations, Math. Control Signals Systems, 26 (2014), 145-176.  doi: 10.1007/s00498-013-0110-x.

[3]

K. BeauchardP. Cannarsa and R. Guglielmi, Null controllability of Grushin-type operators in dimension two, J. Eur. Math. Soc. (JEMS), 16 (2014), 67-101.  doi: 10.4171/JEMS/428.

[4]

K. Beauchard, J. Dardé and S. Ervedoza, Minimal time issues for the observability of Grushin-type equations, December 2017.

[5]

K. BeauchardB. HelfferR. Henry and L. Robbiano, Degenerate parabolic operators of Kolmogorov type with a geometric control condition, ESAIM Control Optim. Calc. Var., 21 (2015), 487-512.  doi: 10.1051/cocv/2014035.

[6]

K. BeauchardL. Miller and M. Morancey, 2D Grushin-type equations: Minimal time and null controllable data, J. Differential Equations, 259 (2015), 5813-5845.  doi: 10.1016/j.jde.2015.07.007.

[7]

K. Beauchard and K. Pravda-Starov, Null-controllability of non-autonomous Ornstein-Uhlenbeck equations, J. Math. Anal. Appl., 456 (2017), 496–524. doi: 10.1016/j.jmaa.2017.07.014.

[8]

L. S. Boulton, Non-self-adjoint harmonic oscillator, compact semigroups and pseudospectra, J. Operator Theory, 47 (2002), 413-429. 

[9] E. B. Davies, Heat Kernels and Spectral Theory, volume 92 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1989.  doi: 10.1017/CBO9780511566158.
[10]

E. B. Davies, Pseudo-spectra, the harmonic oscillator and complex resonances, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 455 (1999), 585-599.  doi: 10.1098/rspa.1999.0325.

[11]

E. B. Davies, Semi-classical states for non-self-adjoint Schrödinger operators, Comm. Math. Phys., 200 (1999), 35-41.  doi: 10.1007/s002200050521.

[12]

E. B. Davies and A. B. J. Kuijlaars, Spectral asymptotics of the non-self-adjoint harmonic oscillator, J. London Math. Soc. (2), 70 (2004), 420-426.  doi: 10.1112/S0024610704005381.

[13]

T. Duyckaerts and L. Miller, Resolvent conditions for the control of parabolic equations, J. Funct. Anal., 263 (2012), 3641-3673.  doi: 10.1016/j.jfa.2012.09.003.

[14]

Ju. V. Egorov, Some problems in the theory of optimal control, Ž. Vyčisl. Mat. i Mat. Fiz., 3 (1963), 887-904. 

[15]

L. EscauriazaC. E. KenigG. Ponce and L. Vega, Hardy's uncertainty principle, convexity and Schrödinger evolutions, J. Eur. Math. Soc. (JEMS), 10 (2008), 883-907.  doi: 10.4171/JEMS/134.

[16]

L. EscauriazaG. Seregin and V. Šverák, Backward uniqueness for the heat operator in half-space, Algebra i Analiz, 15 (2003), 201-214. 

[17]

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

[18]

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

[19]

A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, volume 34 of Lecture Notes Series, Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul, 1996.

[20]

B. Helffer, Semi-classical Analysis for the Schrödinger Operator and Applications, volume 1336 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1988. doi: 10.1007/BFb0078115.

[21]

L. IskauriazaG. A. Serëgin and V. Shverak, $L_{3, \infty}$-solutions of Navier-Stokes equations and backward uniqueness, Uspekhi Mat. Nauk, 58 (2003), 3-44.  doi: 10.1070/RM2003v058n02ABEH000609.

[22]

F. John, Partial Differential Equations, volume 1 of Applied Mathematical Sciences, Springer-Verlag, New York, fourth edition, 1982. doi: 10.1007/978-1-4684-9333-7.

[23]

B. Frank Jones and Jr ., A fundamental solution for the heat equation which is supported in a strip, J. Math. Anal. Appl., 60 (1977), 314-324.  doi: 10.1016/0022-247X(77)90021-X.

[24]

A. Koenig, Non null controllability of the Grushin equation in 2D, C. R. Math. Acad. Sci. Paris, 355 (2017), 1215–1235. https://arXiv.org/abs/1701.06467, 2017. doi: 10.1016/j.crma.2017.10.021.

[25]

C. Laurent and M. Leautaud, Tunneling estimates and approximate controllability for hypoelliptic equations, https://arXiv.org/abs/1703.10797, March 2017.

[26]

J. Le Rousseau and I. Moyano, Null-controllability of the Kolmogorov equation in the whole phase space, J. Differential Equations, 260 (2016), 3193-3233.  doi: 10.1016/j.jde.2015.09.062.

[27]

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.

[28]

B. Ya. Levin, Lectures on Entire Functions, volume 150 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1996. In collaboration with and with a preface by Yu. Lyubarskii, M. Sodin and V. Tkachenko, Translated from the Russian manuscript by Tkachenko.

[29]

L. Li and V. Šverák, Backward uniqueness for the heat equation in cones, Comm. Partial Differential Equations, 37 (2012), 1414-1429.  doi: 10.1080/03605302.2011.635323.

[30]

S. Micu and E. Zuazua, On the lack of null-controllability of the heat equation on the half-line, Trans. Amer. Math. Soc., 353 (2001), 1635–1659 (electronic). doi: 10.1090/S0002-9947-00-02665-9.

[31]

S. Micu and E. Zuazua, On the controllability of a fractional order parabolic equation, SIAM J. Control Optim., 44 (2006), 1950–1972 (electronic). doi: 10.1137/S036301290444263X.

[32]

L. Miller, On the null-controllability of the heat equation in unbounded domains, Bull. Sci. Math., 129 (2005), 175-185.  doi: 10.1016/j.bulsci.2004.04.003.

[33]

L. Miller, Unique continuation estimates for the Laplacian and the heat equation on non-compact manifolds, Math. Res. Lett., 12 (2005), 37-47.  doi: 10.4310/MRL.2005.v12.n1.a4.

[34]

L. Miller, On the controllability of anomalous diffusions generated by the fractional Laplacian, Math. Control Signals Systems, 18 (2006), 260-271.  doi: 10.1007/s00498-006-0003-3.

[35]

L. Miller, Unique continuation estimates for sums of semiclassical eigenfunctions and null-controllability from cones, 20 pages, 1 figure, AMS-LaTeX., November 2008.

[36]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, volume 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[37]

J. Rauch, Partial Differential Equations, volume 128 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0953-9.

[38]

A. Rüland, On the backward uniqueness property for the heat equation in two-dimensional conical domains, Manuscripta Math., 147 (2015), 415-436.  doi: 10.1007/s00229-015-0764-4.

[39]

G. Seregin and V. Šverák, The Navier-Stokes equations and backward uniqueness, In Nonlinear Problems in Mathematical Physics and Related Topics, II, volume 2 of Int. Math. Ser. (N. Y.), pages 353–366. Kluwer/Plenum, New York, 2002.

[40]

G. Wang, M. Wang, C. Zhang and Y. Zhang, Observable set, observability, interpolation inequality and spectral inequality for the heat equation in $\mathbb{R}^n$, J. Math. Pures Appl., 126 (2019), 144–194, https://arXiv.org/abs/1711.04279. doi: 10.1016/j.matpur.2019.04.009.

[41]

D. V. Widder, The role of the Appell transformation in the theory of heat conduction, Trans. Amer. Math. Soc., 109 (1963), 121-134.  doi: 10.1090/S0002-9947-1963-0154068-2.

[42]

J. Wu and W. Wang, On backward uniqueness for the heat operator in cones, J. Differential Equations, 258 (2015), 224-241.  doi: 10.1016/j.jde.2014.09.011.

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