Observability estimate for the parabolic equations with inverse square potential

Guojie Zheng; Baolin Ma; Guojie Zheng; Baolin Ma

doi:10.3934/math.2021785

AIMS Mathematics

2021, Volume 6, Issue 12: 13525-13532. doi: 10.3934/math.2021785

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Research article

Observability estimate for the parabolic equations with inverse square potential

Guojie Zheng ^{1,2
,
,},
Baolin Ma ³

1.
College of Digital Technology and Engineering, Ningbo University of Finance & Economics, Ningbo, 315175, China
2.
College of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, China
3.
School of Mathematical Science, Henan Institute of Science and Technology, Xinxiang, 453003, China

Received: 13 July 2021 Accepted: 13 September 2021 Published: 22 September 2021
MSC : 35K10, 93B07

This paper investigates an observability estimate for the parabolic equations with inverse square potential in a $ C^2 $ bounded domain $ \Omega\subset\mathbb{R}^d $, which contains $ 0 $. The observation region is a product set of a subset $ E\subset(0, T] $ with positive measure and a non-empty open subset $ \omega\subset\Omega $ with $ 0\notin\omega $. We build up this estimate by a delicate result in measure theory in ^[7] and the Lebeau-Robbiano strategy.
- observability estimate,
- parabolic equations,
- spectral inequality,
- inverse square potential
Citation: Guojie Zheng, Baolin Ma. Observability estimate for the parabolic equations with inverse square potential[J]. AIMS Mathematics, 2021, 6(12): 13525-13532. doi: 10.3934/math.2021785

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Abstract

This paper investigates an observability estimate for the parabolic equations with inverse square potential in a $ C^2 $ bounded domain $ \Omega\subset\mathbb{R}^d $, which contains $ 0 $. The observation region is a product set of a subset $ E\subset(0, T] $ with positive measure and a non-empty open subset $ \omega\subset\Omega $ with $ 0\notin\omega $. We build up this estimate by a delicate result in measure theory in ^[7] and the Lebeau-Robbiano strategy.

References

[1]	J. Apraiz, L. Escauriaza, G. Wang, C. Zhang, Observability inequalities and measurable sets, J. Eur. Math. Soc., 16 (2014), 2433–2475. doi: 10.4171/JEMS/490
[2]	P. Baras, J. A. Goldstein, The heat equation with a singular potential, Trans. Amer. Math. Soc., 284 (1984), 121–139. doi: 10.1090/S0002-9947-1984-0742415-3
[3]	S. Ervedoza, Control and stabilization properties for a singular heat equation with an inverse square potential, Commun. Part. Diff. Eq., 33 (2008), 1996–2019. doi: 10.1080/03605300802402633
[4]	J. A. Goldstein, Q. S. Zhang, Linear parabolic equations with strong singular potentials, Trans. Amer. Math. Soc., 355 (2003), 197–211.
[5]	J. L. Lions, Exact controllability, stabilization and perturbations for distributed system, SIAM Rev., 30 (1988), 1–68. doi: 10.1137/1030001
[6]	K. D. Phung, Carleman commutator approach in logarithmic convexity for parabolic equations, Math. Control Relat. F., 8 (2018), 899–933. doi: 10.3934/mcrf.2018040
[7]	K. D. Phung, G. Wang, An observability estimate for the parabolic equations from a measurable set in time and its applications, J. Eur. Math. Soc., 15 (2013), 681–703. doi: 10.4171/JEMS/371
[8]	K. D. Phung, L. Wang, C. Zhang, Bang-bang property for time optimal control of semilinear heat equation, Ann. I. H. Poincare-An., 31 (2014), 477–499. doi: 10.1016/j.anihpc.2013.04.005
[9]	A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983.
[10]	I. Peral, J. L. Vazquez, On the stability or instability of the singular solution of the semilinear heat equation with exponential reaction term, Arch. Ration. Mech. An., 129 (1995), 201–224. doi: 10.1007/BF00383673
[11]	J. L. Vazquez, E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal., 173 (2000), 103–153. doi: 10.1006/jfan.1999.3556
[12]	J. Vancostenoble, 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
[13]	G. Wang, $L^\infty$-null controllability for the heat equation and its consequences for the time optimal control problem, SIAM J. Control Optim., 47 (2008), 1701–1720. doi: 10.1137/060678191
[14]	G. Wang, L. Wang, The Carleman inequality and its application to periodic optimal control governed by semilinear parabolic differential equations, J. Optimz. Theory App., 118 (2003), 429–461. doi: 10.1023/A:1025459624398
[15]	C. Zhang, An observability estimate for the heat equation from a product of two measurable sets, J. Math. Anal. Appl., 396 (2012), 7–12. doi: 10.1016/j.jmaa.2012.05.082
[16]	G. Zheng, K. Li, Y. Zhang, Quantitative unique continuation for the heat equations with inverse square potential, J. Inequal. Appl., 1 (2018), 1–17.

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