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

A quantitative internal unique continuation for stochastic parabolic equations

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.
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.

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