February  2021, 20(2): 547-558. doi: 10.3934/cpaa.2020280

A unique continuation property for a class of parabolic differential inequalities in a bounded domain

1. 

College of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, China

2. 

College of Engineering, Huazhong Agricultural University, Wuhan, 430070, China

3. 

Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221, USA

* Corresponding author

Received  May 2020 Revised  September 2020 Published  December 2020

Fund Project: Guojie Zheng is supported by the Natural Science Foundation of Henan Province (No. 202300410248) and the Natural Science Foundation of Henan Province (No. 2019PL15), and Taige Wang is supported by Faculty Development Fund granted by McMicken College of Arts and Sciences, University of Cincinnati

This article is concerned with a strong unique continuation property of a forward differential inequality abstracted from parabolic equations proposed on a convex domain $ \Omega $ prescribed with some regularity and growth conditions. Our results show that the value of the solutions can be determined uniquely by its value on an arbitrary open subset $ \omega $ in $ \Omega $ at any given positive time $ T $. We also derive the quantitative nature of this unique continuation, that is, the estimate of a $ L^2(\Omega) $ norm of the initial data, which is majorized by that of solution on the bounded open subset $ \omega $ at terminal moment $ t = T $.

Citation: Guojie Zheng, Dihong Xu, Taige Wang. A unique continuation property for a class of parabolic differential inequalities in a bounded domain. Communications on Pure & Applied Analysis, 2021, 20 (2) : 547-558. doi: 10.3934/cpaa.2020280
References:
[1]

M. Bellassoued and M. Yamamoto, Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems, Springer, Tokyo, 2017. doi: 10.1007/978-4-431-56600-7.  Google Scholar

[2]

T. Carleman, Sur un problème d'unicité pur les systèmes d'équations aux dérivées partielles à deux variables indépedantes, Ark. Mat., Astr. Fys., 26 (1939), 9pp.  Google Scholar

[3]

G. Camliyurt and I. Kukavica, Quantitative unique continuation for a parabolic equation, Indiana Univ. Math. J., 67 (2018), 657-678. doi: 10.1512/iumj.2018.67.7283.  Google Scholar

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H. Donnelly and C. Fefferman, Nodal sets of eigenfunctions on Riemann manifolds, Invent. Math., 93 (1988), 161-183. doi: 10.1007/BF01393691.  Google Scholar

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

[6]

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

[7]

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

[8]

N. Garofalo and F. Lin, Monotonicity properties of variational integrals, $A_p$ weights and unique continuation, Indiana Univ. Math. J., 35 (1986), 245-268. doi: 10.1512/iumj.1986.35.35015.  Google Scholar

[9]

D. Jerison and C. E. Kenig, Unique continuation and absence of positive eigenvalues for Schorodinger operators, Ann. Math., 121 (1985), 463-488. doi: 10.2307/1971205.  Google Scholar

[10]

C. E. Kenig, Quantitative unique continuation, logarithmic convexity of Gaussian means and Hardy's uncertainty principle, Proc. Sympos. Pure Math., 79 (2008), 207{227. doi: 10.1090/pspum/079/2500494.  Google Scholar

[11]

I. Kukavica, Quantitative uniqueness for second-order elliptic operators, Duke Math. J., 91 (1998), 225-240. doi: 10.1215/S0012-7094-98-09111-6.  Google Scholar

[12]

H. Koch and D. Tataru, Carleman estimates and unique continuation for second order parabolic equations with nonsmooth coefficients, Commun. PDE, 34 (2009), 305-366. doi: 10.1080/03605300902740395.  Google Scholar

[13]

E. M. Landis and O. A. Oleinik, Generalized analyticity and some related properties of solutions of elliptic and parabolic equations, Russ. Math. Surv+, 29 (1974), 195-212.  Google Scholar

[14]

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

[15]

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.  Google Scholar

[16]

C. Poon, Unique continuation for parabolic equations, Comm. PDE, 21 (1996), 521-539. doi: 10.1080/03605309608821195.  Google Scholar

[17]

J. C. Saut and E. Scheurer, Unique continuation for evolution equations, J. Differ. Equ., 66 (1987), 118-137. doi: 10.1016/0022-0396(87)90043-X.  Google Scholar

[18]

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

[19]

M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Prob., 25 (2009), 123013. doi: 10.1088/0266-5611/25/12/123013.  Google Scholar

show all references

References:
[1]

M. Bellassoued and M. Yamamoto, Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems, Springer, Tokyo, 2017. doi: 10.1007/978-4-431-56600-7.  Google Scholar

[2]

T. Carleman, Sur un problème d'unicité pur les systèmes d'équations aux dérivées partielles à deux variables indépedantes, Ark. Mat., Astr. Fys., 26 (1939), 9pp.  Google Scholar

[3]

G. Camliyurt and I. Kukavica, Quantitative unique continuation for a parabolic equation, Indiana Univ. Math. J., 67 (2018), 657-678. doi: 10.1512/iumj.2018.67.7283.  Google Scholar

[4]

H. Donnelly and C. Fefferman, Nodal sets of eigenfunctions on Riemann manifolds, Invent. Math., 93 (1988), 161-183. doi: 10.1007/BF01393691.  Google Scholar

[5]

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

[6]

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

[7]

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

[8]

N. Garofalo and F. Lin, Monotonicity properties of variational integrals, $A_p$ weights and unique continuation, Indiana Univ. Math. J., 35 (1986), 245-268. doi: 10.1512/iumj.1986.35.35015.  Google Scholar

[9]

D. Jerison and C. E. Kenig, Unique continuation and absence of positive eigenvalues for Schorodinger operators, Ann. Math., 121 (1985), 463-488. doi: 10.2307/1971205.  Google Scholar

[10]

C. E. Kenig, Quantitative unique continuation, logarithmic convexity of Gaussian means and Hardy's uncertainty principle, Proc. Sympos. Pure Math., 79 (2008), 207{227. doi: 10.1090/pspum/079/2500494.  Google Scholar

[11]

I. Kukavica, Quantitative uniqueness for second-order elliptic operators, Duke Math. J., 91 (1998), 225-240. doi: 10.1215/S0012-7094-98-09111-6.  Google Scholar

[12]

H. Koch and D. Tataru, Carleman estimates and unique continuation for second order parabolic equations with nonsmooth coefficients, Commun. PDE, 34 (2009), 305-366. doi: 10.1080/03605300902740395.  Google Scholar

[13]

E. M. Landis and O. A. Oleinik, Generalized analyticity and some related properties of solutions of elliptic and parabolic equations, Russ. Math. Surv+, 29 (1974), 195-212.  Google Scholar

[14]

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

[15]

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.  Google Scholar

[16]

C. Poon, Unique continuation for parabolic equations, Comm. PDE, 21 (1996), 521-539. doi: 10.1080/03605309608821195.  Google Scholar

[17]

J. C. Saut and E. Scheurer, Unique continuation for evolution equations, J. Differ. Equ., 66 (1987), 118-137. doi: 10.1016/0022-0396(87)90043-X.  Google Scholar

[18]

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

[19]

M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Prob., 25 (2009), 123013. doi: 10.1088/0266-5611/25/12/123013.  Google Scholar

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