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 and 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), 113-127. doi: 10.1215/S0012-7094-00-10415-2.

[2]

L. Escauriaza and L. Vega, Carleman inequalities and the heat operator II, Indiana U. Math. J., 50 (2001), 1149-1169. doi: 10.1512/iumj.2001.50.1937.

[3]

V. Isakov, Carleman type estimates in an anisotropic case and applications, J. Differential Equations, 105 (1993), 217-238. doi: 10.1006/jdeq.1993.1088.

[4]

H. Li and Q. Lü, A quantitative boundary unique continuation for stochastic parabolic equations, J. Math. Anal. Appl., 402 (2013), 518-526. doi: 10.1016/j.jmaa.2013.01.038.

[5]

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

[6]

Q. Lü, Observability estimate for stochastic Schrödinger equations and its applications, SIAM J. Control Optim., 51 (2013), 121-144. doi: 10.1137/110830964.

[7]

Q. Lü, Observability estimate and state observation problems for stochastic hyperbolic equations, Inverse Problems, 29 (2013), 095011, 22 pp. doi: 10.1088/0266-5611/29/9/095011.

[8]

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

[9]

J.-C. Saut and B. Scheurer, Unique continuation for soome evolution equations, J. Differential Equations, 66 (1987), 118-139. doi: 10.1016/0022-0396(87)90043-X.

[10]

C. D. Sogge, A unique continuation theorem for second order parabolic differential operators, Ark. Mat., 28 (1990), 159-182. doi: 10.1007/BF02387373.

[11]

S. Tang and X. Zhang, Null controllability for forward and backward stochastic parabolic equations, SIAM J. Control Optim., 48 (2009), 2191-2216. doi: 10.1137/050641508.

[12]

H. Yamabe, A unique continuation theorem of a diffusion equation, Ann. Math., 69 (1959), 462-466. doi: 10.2307/1970194.

[13]

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

[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, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2007, 527-621. doi: 10.1016/S1874-5717(07)80010-7.

[15]

C. Zuily, Uniqueness and Non-Uniqueness in the Cauchy Problem, Progress in Mathematics, 33, Birkhäuser, Boston, 1983. doi: 10.1007/978-1-4899-6656-8.

show all references

References:
[1]

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

[2]

L. Escauriaza and L. Vega, Carleman inequalities and the heat operator II, Indiana U. Math. J., 50 (2001), 1149-1169. doi: 10.1512/iumj.2001.50.1937.

[3]

V. Isakov, Carleman type estimates in an anisotropic case and applications, J. Differential Equations, 105 (1993), 217-238. doi: 10.1006/jdeq.1993.1088.

[4]

H. Li and Q. Lü, A quantitative boundary unique continuation for stochastic parabolic equations, J. Math. Anal. Appl., 402 (2013), 518-526. doi: 10.1016/j.jmaa.2013.01.038.

[5]

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

[6]

Q. Lü, Observability estimate for stochastic Schrödinger equations and its applications, SIAM J. Control Optim., 51 (2013), 121-144. doi: 10.1137/110830964.

[7]

Q. Lü, Observability estimate and state observation problems for stochastic hyperbolic equations, Inverse Problems, 29 (2013), 095011, 22 pp. doi: 10.1088/0266-5611/29/9/095011.

[8]

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

[9]

J.-C. Saut and B. Scheurer, Unique continuation for soome evolution equations, J. Differential Equations, 66 (1987), 118-139. doi: 10.1016/0022-0396(87)90043-X.

[10]

C. D. Sogge, A unique continuation theorem for second order parabolic differential operators, Ark. Mat., 28 (1990), 159-182. doi: 10.1007/BF02387373.

[11]

S. Tang and X. Zhang, Null controllability for forward and backward stochastic parabolic equations, SIAM J. Control Optim., 48 (2009), 2191-2216. doi: 10.1137/050641508.

[12]

H. Yamabe, A unique continuation theorem of a diffusion equation, Ann. Math., 69 (1959), 462-466. doi: 10.2307/1970194.

[13]

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

[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, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2007, 527-621. doi: 10.1016/S1874-5717(07)80010-7.

[15]

C. Zuily, Uniqueness and Non-Uniqueness in the Cauchy Problem, Progress in Mathematics, 33, Birkhäuser, Boston, 1983. doi: 10.1007/978-1-4899-6656-8.

[1]

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

Guojie Zheng, Dihong Xu, Taige Wang. A unique continuation property for a class of parabolic differential inequalities in a bounded domain. Communications on Pure and Applied Analysis, 2021, 20 (2) : 547-558. doi: 10.3934/cpaa.2020280
[3]

Peng Gao. Carleman estimates and Unique Continuation Property for 1-D viscous Camassa-Holm equation. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 169-188. doi: 10.3934/dcds.2017007
[4]

Agnid Banerjee. A note on the unique continuation property for fully nonlinear elliptic equations. Communications on Pure and Applied Analysis, 2015, 14 (2) : 623-626. doi: 10.3934/cpaa.2015.14.623
[5]

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

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

Gunther Uhlmann, Jenn-Nan Wang. Unique continuation property for the elasticity with general residual stress. Inverse Problems and Imaging, 2009, 3 (2) : 309-317. doi: 10.3934/ipi.2009.3.309
[8]

Ihyeok Seo. Carleman estimates for the Schrödinger operator and applications to unique continuation. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1013-1036. doi: 10.3934/cpaa.2012.11.1013
[9]

Peng Gao. Global Carleman estimate for the Kawahara equation and its applications. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1853-1874. doi: 10.3934/cpaa.2018088
[10]

Taige Wang, Dihong Xu. A quantitative strong unique continuation property of a diffusive SIS model. Discrete and Continuous Dynamical Systems - S, 2022, 15 (6) : 1599-1614. doi: 10.3934/dcdss.2022024
[11]

A. Alexandrou Himonas, Gerard Misiołek, Feride Tiǧlay. On unique continuation for the modified Euler-Poisson equations. Discrete and Continuous Dynamical Systems, 2007, 19 (3) : 515-529. doi: 10.3934/dcds.2007.19.515
[12]

Agnid Banerjee, Ramesh Manna. Carleman estimates for a class of variable coefficient degenerate elliptic operators with applications to unique continuation. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5105-5139. doi: 10.3934/dcds.2021070
[13]

Lucie Baudouin, Emmanuelle Crépeau, Julie Valein. Global Carleman estimate on a network for the wave equation and application to an inverse problem. Mathematical Control and Related Fields, 2011, 1 (3) : 307-330. doi: 10.3934/mcrf.2011.1.307
[14]

Giovanni Covi, Keijo Mönkkönen, Jesse Railo. Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems. Inverse Problems and Imaging, 2021, 15 (4) : 641-681. doi: 10.3934/ipi.2021009
[15]

Kim Dang Phung. Carleman commutator approach in logarithmic convexity for parabolic equations. Mathematical Control and Related Fields, 2018, 8 (3&4) : 899-933. doi: 10.3934/mcrf.2018040
[16]

Mehdi Badra. Global Carleman inequalities for Stokes and penalized Stokes equations. Mathematical Control and Related Fields, 2011, 1 (2) : 149-175. doi: 10.3934/mcrf.2011.1.149
[17]

Yanpeng Jin, Ying Fu. Global Carleman estimate and its applications for a sixth-order equation related to thin solid films. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022072
[18]

Jong-Shenq Guo, Satoshi Sasayama, Chi-Jen Wang. Blowup rate estimate for a system of semilinear parabolic equations. Communications on Pure and Applied Analysis, 2009, 8 (2) : 711-718. doi: 10.3934/cpaa.2009.8.711
[19]

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
[20]

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

2020 Impact Factor: 1.284

Tools

Metrics

  • PDF downloads (143)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy