American Institute of Mathematical Sciences

Advanced
March  2015, 5(1): 165-176. doi: 10.3934/mcrf.2015.5.165

A quantitative internal unique continuation for stochastic parabolic equations

Zhongqi Yin 1,

1. 

School of Mathematics, Sichuan Normal University, Chengdu, 610068, China

Received  January 2014 Revised  August 2014 Published  January 2015

This paper is addressed to a quantitative internal unique continuation property for stochastic parabolic equations, i.e., we show that each of their solutions can be determined by the observation on any nonempty open subset of the whole region in which the equations evolve. The proof is based on a global Carleman estimate.
Keywords: global Carleman estimate., internal unique continuation property, Stochastic parabolic equations.
Mathematics Subject Classification: Primary: 60H15; Secondary: 34A1.
Citation: Zhongqi Yin. A quantitative internal unique continuation for stochastic parabolic equations. Mathematical Control & Related Fields, 2015, 5 (1) : 165-176. doi: 10.3934/mcrf.2015.5.165
References:
[1]

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

[2]

L. Escauriaza and L. Vega, Carleman inequalities and the heat operator II,, Indiana U. Math. J., 50 (2001), 1149.  doi: 10.1512/iumj.2001.50.1937.  Google Scholar

[3]

V. Isakov, Carleman type estimates in an anisotropic case and applications,, J. Differential Equations, 105 (1993), 217.  doi: 10.1006/jdeq.1993.1088.  Google Scholar

[4]

H. Li and Q. Lü, A quantitative boundary unique continuation for stochastic parabolic equations,, J. Math. Anal. Appl., 402 (2013), 518.  doi: 10.1016/j.jmaa.2013.01.038.  Google Scholar

[5]

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

[6]

Q. Lü, Observability estimate for stochastic Schrödinger equations and its applications,, SIAM J. Control Optim., 51 (2013), 121.  doi: 10.1137/110830964.  Google Scholar

[7]

Q. Lü, Observability estimate and state observation problems for stochastic hyperbolic equations,, Inverse Problems, 29 (2013).  doi: 10.1088/0266-5611/29/9/095011.  Google Scholar

[8]

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

[9]

J.-C. Saut and B. Scheurer, Unique continuation for soome evolution equations,, J. Differential Equations, 66 (1987), 118.  doi: 10.1016/0022-0396(87)90043-X.  Google Scholar

[10]

C. D. Sogge, A unique continuation theorem for second order parabolic differential operators,, Ark. Mat., 28 (1990), 159.  doi: 10.1007/BF02387373.  Google Scholar

[11]

S. Tang and X. Zhang, Null controllability for forward and backward stochastic parabolic equations,, SIAM J. Control Optim., 48 (2009), 2191.  doi: 10.1137/050641508.  Google Scholar

[12]

H. Yamabe, A unique continuation theorem of a diffusion equation,, Ann. Math., 69 (1959), 462.  doi: 10.2307/1970194.  Google Scholar

[13]

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

[14]

E. Zuazua, Controllability and observability of partial differential equations: Some results and open problems,, in Handbook of Differential Equations: Evolutionary Differential Equations. Vol. III, (2007), 527.  doi: 10.1016/S1874-5717(07)80010-7.  Google Scholar

[15]

C. Zuily, Uniqueness and Non-Uniqueness in the Cauchy Problem,, Progress in Mathematics, (1983).  doi: 10.1007/978-1-4899-6656-8.  Google Scholar

show all references

References:
[1]

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

[2]

L. Escauriaza and L. Vega, Carleman inequalities and the heat operator II,, Indiana U. Math. J., 50 (2001), 1149.  doi: 10.1512/iumj.2001.50.1937.  Google Scholar

[3]

V. Isakov, Carleman type estimates in an anisotropic case and applications,, J. Differential Equations, 105 (1993), 217.  doi: 10.1006/jdeq.1993.1088.  Google Scholar

[4]

H. Li and Q. Lü, A quantitative boundary unique continuation for stochastic parabolic equations,, J. Math. Anal. Appl., 402 (2013), 518.  doi: 10.1016/j.jmaa.2013.01.038.  Google Scholar

[5]

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

[6]

Q. Lü, Observability estimate for stochastic Schrödinger equations and its applications,, SIAM J. Control Optim., 51 (2013), 121.  doi: 10.1137/110830964.  Google Scholar

[7]

Q. Lü, Observability estimate and state observation problems for stochastic hyperbolic equations,, Inverse Problems, 29 (2013).  doi: 10.1088/0266-5611/29/9/095011.  Google Scholar

[8]

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

[9]

J.-C. Saut and B. Scheurer, Unique continuation for soome evolution equations,, J. Differential Equations, 66 (1987), 118.  doi: 10.1016/0022-0396(87)90043-X.  Google Scholar

[10]

C. D. Sogge, A unique continuation theorem for second order parabolic differential operators,, Ark. Mat., 28 (1990), 159.  doi: 10.1007/BF02387373.  Google Scholar

[11]

S. Tang and X. Zhang, Null controllability for forward and backward stochastic parabolic equations,, SIAM J. Control Optim., 48 (2009), 2191.  doi: 10.1137/050641508.  Google Scholar

[12]

H. Yamabe, A unique continuation theorem of a diffusion equation,, Ann. Math., 69 (1959), 462.  doi: 10.2307/1970194.  Google Scholar

[13]

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

[14]

E. Zuazua, Controllability and observability of partial differential equations: Some results and open problems,, in Handbook of Differential Equations: Evolutionary Differential Equations. Vol. III, (2007), 527.  doi: 10.1016/S1874-5717(07)80010-7.  Google Scholar

[15]

C. Zuily, Uniqueness and Non-Uniqueness in the Cauchy Problem,, Progress in Mathematics, (1983).  doi: 10.1007/978-1-4899-6656-8.  Google Scholar

[1]

Peng Gao. Unique continuation property for stochastic nonclassical diffusion equations and stochastic linearized Benjamin-Bona-Mahony equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2493-2510. doi: 10.3934/dcdsb.2018262
[2]

Peng Gao. Carleman estimates and Unique Continuation Property for 1-D viscous Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 169-188. doi: 10.3934/dcds.2017007
[3]

Agnid Banerjee. A note on the unique continuation property for fully nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2015, 14 (2) : 623-626. doi: 10.3934/cpaa.2015.14.623
[4]

Muriel Boulakia. Quantification of the unique continuation property for the nonstationary Stokes problem. Mathematical Control & Related Fields, 2016, 6 (1) : 27-52. doi: 10.3934/mcrf.2016.6.27
[5]

Laurent Bourgeois. Quantification of the unique continuation property for the heat equation. Mathematical Control & Related Fields, 2017, 7 (3) : 347-367. doi: 10.3934/mcrf.2017012
[6]

Gunther Uhlmann, Jenn-Nan Wang. Unique continuation property for the elasticity with general residual stress. Inverse Problems & Imaging, 2009, 3 (2) : 309-317. doi: 10.3934/ipi.2009.3.309
[7]

Ihyeok Seo. Carleman estimates for the Schrödinger operator and applications to unique continuation. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1013-1036. doi: 10.3934/cpaa.2012.11.1013
[8]

Peng Gao. Global Carleman estimate for the Kawahara equation and its applications. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1853-1874. doi: 10.3934/cpaa.2018088
[9]

A. Alexandrou Himonas, Gerard Misiołek, Feride Tiǧlay. On unique continuation for the modified Euler-Poisson equations. Discrete & Continuous Dynamical Systems - A, 2007, 19 (3) : 515-529. doi: 10.3934/dcds.2007.19.515
[10]

Lucie Baudouin, Emmanuelle Crépeau, Julie Valein. Global Carleman estimate on a network for the wave equation and application to an inverse problem. Mathematical Control & Related Fields, 2011, 1 (3) : 307-330. doi: 10.3934/mcrf.2011.1.307
[11]

Kim Dang Phung. Carleman commutator approach in logarithmic convexity for parabolic equations. Mathematical Control & Related Fields, 2018, 8 (3&4) : 899-933. doi: 10.3934/mcrf.2018040
[12]

Mehdi Badra. Global Carleman inequalities for Stokes and penalized Stokes equations. Mathematical Control & Related Fields, 2011, 1 (2) : 149-175. doi: 10.3934/mcrf.2011.1.149
[13]

Jong-Shenq Guo, Satoshi Sasayama, Chi-Jen Wang. Blowup rate estimate for a system of semilinear parabolic equations. Communications on Pure & Applied Analysis, 2009, 8 (2) : 711-718. doi: 10.3934/cpaa.2009.8.711
[14]

Gary Lieberman. A new regularity estimate for solutions of singular parabolic equations. Conference Publications, 2005, 2005 (Special) : 605-610. doi: 10.3934/proc.2005.2005.605
[15]

José G. Llorente. Mean value properties and unique continuation. Communications on Pure & Applied Analysis, 2015, 14 (1) : 185-199. doi: 10.3934/cpaa.2015.14.185
[16]

B. E. Ainseba, Sebastian Aniţa. Internal nonnegative stabilization for some parabolic equations. Communications on Pure & Applied Analysis, 2008, 7 (3) : 491-512. doi: 10.3934/cpaa.2008.7.491
[17]

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
[18]

Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367
[19]

Chunpeng Wang, Yanan Zhou, Runmei Du, Qiang Liu. Carleman estimate for solutions to a degenerate convection-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4207-4222. doi: 10.3934/dcdsb.2018133
[20]

Genni Fragnelli. Null controllability of degenerate parabolic equations in non divergence form via Carleman estimates. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 687-701. doi: 10.3934/dcdss.2013.6.687

2018 Impact Factor: 1.292

Tools

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences