September  2018, 8(3&4): 1097-1116. doi: 10.3934/mcrf.2018047

Quantitative unique continuation for the heat equation with Coulomb potentials

1. 

School of Mathematics and Statistics, Wuhan University, 430072 Wuhan, China

2. 

Department of Mathematics, University of the Basque Country (UPV/EHU), 48940 Leioa, Bilbao, Spain

* Corresponding author: Can Zhang

Received  July 2017 Revised  September 2017 Published  September 2018

Fund Project: The author is supported by the National Natural Science Foundation of China under grants 11501424, and by Ministerio de Ciencia e Innovación grant MTM2014-53145-P, Spain.

In this paper, we establish a Hölder-type quantitative estimate of unique continuation for solutions to the heat equation with Coulomb potentials in either a bounded convex domain or a $C^2$-smooth bounded domain. The approach is based on the frequency function method, as well as some parabolic-type Hardy inequalities.

Citation: Can Zhang. Quantitative unique continuation for the heat equation with Coulomb potentials. Mathematical Control & Related Fields, 2018, 8 (3&4) : 1097-1116. doi: 10.3934/mcrf.2018047
References:
[1]

J. ApraizL. EscauriazaG. Wang and C. Zhang, Observability inequalities and measurable sets, J. Eur. Math. Soc., 16 (2014), 2433-2475.  doi: 10.4171/JEMS/490.  Google Scholar

[2]

C. Bardos and K. D. Phung, Observation estimate for kinetic transport equation by diffusion approximation, Comptes Rendus Mathematique, 355 (2017), 640-664.  doi: 10.1016/j.crma.2017.04.017.  Google Scholar

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X. Y. Chen, A strong unique continuation theorem for parabolic equations, Mathematische Annalen, 311 (1998), 603-630.  doi: 10.1007/s002080050202.  Google Scholar

[4]

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

[5]

L. EscauriazaS. Montaner and C. Zhang, Observation from measurable sets for parabolic analytic evolutions and applications, J. Math. Pures Appl., 104 (2015), 837-867.  doi: 10.1016/j.matpur.2015.05.005.  Google Scholar

[6]

L. EscauriazaS. Montaner and C. Zhang, Analyticity of solutions to parabolic evolutions and applications, SIAM Journal on Mathematical Analysis, 49 (2017), 4064-4092.  doi: 10.1137/15M1039705.  Google Scholar

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L. Escauriaza, Carleman inequalities and the heat operator, Duke Math. J., 104 (2000), 113-127.  doi: 10.1215/S0012-7094-00-10415-2.  Google Scholar

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L. Escauriaza and F. J. Fernández, Unique continuation for parabolic operators, Ark. Mat., 41 (2003), 35-60.  doi: 10.1007/BF02384566.  Google Scholar

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L. EscauriazaF. J. Fernández and S. Vessella, Doubling properties of caloric functions, Appl. Anal., 85 (2006), 205-223.  doi: 10.1080/00036810500277082.  Google Scholar

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L. C. Evans, Partial Differential Equations, Volume 19 of Graduate Studies in Mathematics, American Mathematical Soc., 2010. doi: 10.1090/gsm/019.  Google Scholar

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V. Felli and A. Primo, Classification of local asymptotic for solutions to heat equations with inverse-square potentials, Discrete and Continuous Dynamical Systems - A, 31 (2011), 65-107.  doi: 10.3934/dcds.2011.31.65.  Google Scholar

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F. J. Fernández, Unique continuation for parabolic operators Ⅱ, Communications in Partial Differential Equations, 28 (2003), 1597-1604.  doi: 10.1081/PDE-120024523.  Google Scholar

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N. Garofalo and F. H. Lin, Monotonicity properties of variational integrals: Ap weights and unique continuation, Indiana University Math. J., 35 (1986), 245-268.  doi: 10.1512/iumj.1986.35.35015.  Google Scholar

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N. Garofalo and F. H. Lin, Unique continuation for elliptic operators: A geometric-variation approach, Comm. Pure. Appl. Math, 40 (1987), 347-366.  doi: 10.1002/cpa.3160400305.  Google Scholar

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I. Kukavica and K. Nyström, Unique continuation on the boundary for Dini domains, Proc. Amer. Math. Soc., 126 (1998), 441-446.  doi: 10.1090/S0002-9939-98-04065-9.  Google Scholar

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X. Li and J. Yong, Optimal Control Theory for Infinite-Dimensional Systems, Systems & Control: Foundations & Applications, Inc., Boston, MA, 1995. doi: 10.1007/978-1-4612-4260-4.  Google Scholar

[17]

F. H. Lin, A uniqueness theorem for parabolic equations, Comm. Pure. Appl. Math, 43 (1990), 127-136.  doi: 10.1002/cpa.3160430105.  Google Scholar

[18]

J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin Heildeberg New York, 1971.  Google Scholar

[19]

Q. Lü and Z. Yin, Unique continuation for stochastic heat equations, ESAIM Control Optim. Calc. Var., 21 (2015), 378-398.  doi: 10.1051/cocv/2014027.  Google Scholar

[20]

Q. Lü, Strong unique continuation property for stochastic parabolic equations, preprint, arXiv: 1701.02136. Google Scholar

[21]

T. Okaji, A note on unique continuation for parabolic operators with singular potentials, in Studies in Phase Space Analysis with Applications to PDEs, Springer New York, 84 (2013), 291-312. doi: 10.1007/978-1-4614-6348-1_13.  Google Scholar

[22]

K. D. Phung, Carleman commutator approach in logarithmic convexity for parabolic equations, preprint. Google Scholar

[23]

K. D. Phung and G. Wang, Quantitative unique continuation for the semilinear heat equation in a convex domain, J. Funct. Anal., 259 (2010), 1230-1247.  doi: 10.1016/j.jfa.2010.04.015.  Google Scholar

[24]

K. D. Phung and G. Wang, An observability estimate for parabolic equations from a measurable set in time and its applications, J. Eur. Math. Soc., 15 (2013), 681-703.  doi: 10.4171/JEMS/371.  Google Scholar

[25]

K. D. PhungG. Wang and Y. Xu, Impulse output rapid stabilization for heat equations, Journal of Differential Equations, 263 (2017), 5012-5041.  doi: 10.1016/j.jde.2017.06.008.  Google Scholar

[26]

K. D. PhungL. Wang and C. Zhang, Bang-bang property for time optimal control of semilinear heat equation, Annales de I'Institut Henri Poincare (C) Non Linear Analysis, 31 (2014), 477-499.  doi: 10.1016/j.anihpc.2013.04.005.  Google Scholar

[27]

C. C. Poon, Unique continuation for parabolic equations, Communications in Partial Differential Equations, 21 (1996), 521-539.  doi: 10.1080/03605309608821195.  Google Scholar

[28]

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

[29]

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

[30]

S. Vessella, Unique continuation properties and quantitative estimates of unique continuation for parabolic equations, in Handbook of Differential Equations: Evolutionary Equations, 5 (2009), 421-500. doi: 10.1016/S1874-5717(08)00212-0.  Google Scholar

[31]

G. Wang and C. Zhang, Observability inequalities from measurable sets for some abstract evolution equations, SIAM J. Control Optim., 55 (2017), 1862-1886.  doi: 10.1137/15M1051907.  Google Scholar

[32]

G. Wang and E. Zuazua, On the equivalence of minimal time and minimal norm controls for internally controlled heat equations, SIAM J. Control Optim., 50 (2012), 2938-2958.  doi: 10.1137/110857398.  Google Scholar

[33]

L. Wang and Q. Yan, Bang-bang property of time optimal null controls for some semilinear heat equation, SIAM J. Control Optim., 54 (2016), 2949-2964.  doi: 10.1137/140997452.  Google Scholar

[34]

H. Yu, Approximation of time optimal controls for heat equations with perturbations in the system potential, SIAM J. Control Optim., 52 (2014), 1663-1692.  doi: 10.1137/120904251.  Google Scholar

[35]

X. Zhang, Unique continuation for stochastic parabolic equations, Differential Integral Equations, 21 (2008), 81-93.   Google Scholar

[36]

Y. Zhang, Two equivalence theorems of different kinds of optimal control problems for Schrödinger equations, SIAM J. Control Optim., 53 (2015), 926-947.  doi: 10.1137/130941195.  Google Scholar

[37]

Y. Zhang, Unique continuation estimates for the Kolmogorov equation in the whole space, C. R. Math. Acad. Sci. Paris, 354 (2016), 389-393.  doi: 10.1016/j.crma.2016.01.009.  Google Scholar

show all references

References:
[1]

J. ApraizL. EscauriazaG. Wang and C. Zhang, Observability inequalities and measurable sets, J. Eur. Math. Soc., 16 (2014), 2433-2475.  doi: 10.4171/JEMS/490.  Google Scholar

[2]

C. Bardos and K. D. Phung, Observation estimate for kinetic transport equation by diffusion approximation, Comptes Rendus Mathematique, 355 (2017), 640-664.  doi: 10.1016/j.crma.2017.04.017.  Google Scholar

[3]

X. Y. Chen, A strong unique continuation theorem for parabolic equations, Mathematische Annalen, 311 (1998), 603-630.  doi: 10.1007/s002080050202.  Google Scholar

[4]

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

[5]

L. EscauriazaS. Montaner and C. Zhang, Observation from measurable sets for parabolic analytic evolutions and applications, J. Math. Pures Appl., 104 (2015), 837-867.  doi: 10.1016/j.matpur.2015.05.005.  Google Scholar

[6]

L. EscauriazaS. Montaner and C. Zhang, Analyticity of solutions to parabolic evolutions and applications, SIAM Journal on Mathematical Analysis, 49 (2017), 4064-4092.  doi: 10.1137/15M1039705.  Google Scholar

[7]

L. Escauriaza, Carleman inequalities and the heat operator, Duke Math. J., 104 (2000), 113-127.  doi: 10.1215/S0012-7094-00-10415-2.  Google Scholar

[8]

L. Escauriaza and F. J. Fernández, Unique continuation for parabolic operators, Ark. Mat., 41 (2003), 35-60.  doi: 10.1007/BF02384566.  Google Scholar

[9]

L. EscauriazaF. J. Fernández and S. Vessella, Doubling properties of caloric functions, Appl. Anal., 85 (2006), 205-223.  doi: 10.1080/00036810500277082.  Google Scholar

[10]

L. C. Evans, Partial Differential Equations, Volume 19 of Graduate Studies in Mathematics, American Mathematical Soc., 2010. doi: 10.1090/gsm/019.  Google Scholar

[11]

V. Felli and A. Primo, Classification of local asymptotic for solutions to heat equations with inverse-square potentials, Discrete and Continuous Dynamical Systems - A, 31 (2011), 65-107.  doi: 10.3934/dcds.2011.31.65.  Google Scholar

[12]

F. J. Fernández, Unique continuation for parabolic operators Ⅱ, Communications in Partial Differential Equations, 28 (2003), 1597-1604.  doi: 10.1081/PDE-120024523.  Google Scholar

[13]

N. Garofalo and F. H. Lin, Monotonicity properties of variational integrals: Ap weights and unique continuation, Indiana University Math. J., 35 (1986), 245-268.  doi: 10.1512/iumj.1986.35.35015.  Google Scholar

[14]

N. Garofalo and F. H. Lin, Unique continuation for elliptic operators: A geometric-variation approach, Comm. Pure. Appl. Math, 40 (1987), 347-366.  doi: 10.1002/cpa.3160400305.  Google Scholar

[15]

I. Kukavica and K. Nyström, Unique continuation on the boundary for Dini domains, Proc. Amer. Math. Soc., 126 (1998), 441-446.  doi: 10.1090/S0002-9939-98-04065-9.  Google Scholar

[16]

X. Li and J. Yong, Optimal Control Theory for Infinite-Dimensional Systems, Systems & Control: Foundations & Applications, Inc., Boston, MA, 1995. doi: 10.1007/978-1-4612-4260-4.  Google Scholar

[17]

F. H. Lin, A uniqueness theorem for parabolic equations, Comm. Pure. Appl. Math, 43 (1990), 127-136.  doi: 10.1002/cpa.3160430105.  Google Scholar

[18]

J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin Heildeberg New York, 1971.  Google Scholar

[19]

Q. Lü and Z. Yin, Unique continuation for stochastic heat equations, ESAIM Control Optim. Calc. Var., 21 (2015), 378-398.  doi: 10.1051/cocv/2014027.  Google Scholar

[20]

Q. Lü, Strong unique continuation property for stochastic parabolic equations, preprint, arXiv: 1701.02136. Google Scholar

[21]

T. Okaji, A note on unique continuation for parabolic operators with singular potentials, in Studies in Phase Space Analysis with Applications to PDEs, Springer New York, 84 (2013), 291-312. doi: 10.1007/978-1-4614-6348-1_13.  Google Scholar

[22]

K. D. Phung, Carleman commutator approach in logarithmic convexity for parabolic equations, preprint. Google Scholar

[23]

K. D. Phung and G. Wang, Quantitative unique continuation for the semilinear heat equation in a convex domain, J. Funct. Anal., 259 (2010), 1230-1247.  doi: 10.1016/j.jfa.2010.04.015.  Google Scholar

[24]

K. D. Phung and G. Wang, An observability estimate for parabolic equations from a measurable set in time and its applications, J. Eur. Math. Soc., 15 (2013), 681-703.  doi: 10.4171/JEMS/371.  Google Scholar

[25]

K. D. PhungG. Wang and Y. Xu, Impulse output rapid stabilization for heat equations, Journal of Differential Equations, 263 (2017), 5012-5041.  doi: 10.1016/j.jde.2017.06.008.  Google Scholar

[26]

K. D. PhungL. Wang and C. Zhang, Bang-bang property for time optimal control of semilinear heat equation, Annales de I'Institut Henri Poincare (C) Non Linear Analysis, 31 (2014), 477-499.  doi: 10.1016/j.anihpc.2013.04.005.  Google Scholar

[27]

C. C. Poon, Unique continuation for parabolic equations, Communications in Partial Differential Equations, 21 (1996), 521-539.  doi: 10.1080/03605309608821195.  Google Scholar

[28]

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

[29]

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

[30]

S. Vessella, Unique continuation properties and quantitative estimates of unique continuation for parabolic equations, in Handbook of Differential Equations: Evolutionary Equations, 5 (2009), 421-500. doi: 10.1016/S1874-5717(08)00212-0.  Google Scholar

[31]

G. Wang and C. Zhang, Observability inequalities from measurable sets for some abstract evolution equations, SIAM J. Control Optim., 55 (2017), 1862-1886.  doi: 10.1137/15M1051907.  Google Scholar

[32]

G. Wang and E. Zuazua, On the equivalence of minimal time and minimal norm controls for internally controlled heat equations, SIAM J. Control Optim., 50 (2012), 2938-2958.  doi: 10.1137/110857398.  Google Scholar

[33]

L. Wang and Q. Yan, Bang-bang property of time optimal null controls for some semilinear heat equation, SIAM J. Control Optim., 54 (2016), 2949-2964.  doi: 10.1137/140997452.  Google Scholar

[34]

H. Yu, Approximation of time optimal controls for heat equations with perturbations in the system potential, SIAM J. Control Optim., 52 (2014), 1663-1692.  doi: 10.1137/120904251.  Google Scholar

[35]

X. Zhang, Unique continuation for stochastic parabolic equations, Differential Integral Equations, 21 (2008), 81-93.   Google Scholar

[36]

Y. Zhang, Two equivalence theorems of different kinds of optimal control problems for Schrödinger equations, SIAM J. Control Optim., 53 (2015), 926-947.  doi: 10.1137/130941195.  Google Scholar

[37]

Y. Zhang, Unique continuation estimates for the Kolmogorov equation in the whole space, C. R. Math. Acad. Sci. Paris, 354 (2016), 389-393.  doi: 10.1016/j.crma.2016.01.009.  Google Scholar

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