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.

[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.

[3]

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

[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.

[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.

[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.

[7]

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

[8]

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

[9]

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

[10]

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

[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.

[12]

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

[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.

[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.

[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.

[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.

[17]

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

[18]

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

[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.

[20]

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

[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.

[22]

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

[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.

[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.

[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.

[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.

[27]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[35]

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

[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.

[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.

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.

[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.

[3]

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

[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.

[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.

[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.

[7]

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

[8]

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

[9]

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

[10]

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

[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.

[12]

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

[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.

[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.

[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.

[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.

[17]

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

[18]

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

[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.

[20]

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

[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.

[22]

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

[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.

[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.

[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.

[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.

[27]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[35]

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

[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.

[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.

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