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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 |
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 $.
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. |
[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. |
[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. |
[4] |
H. Donnelly and C. Fefferman, Nodal sets of eigenfunctions on Riemann manifolds, Invent. Math., 93 (1988), 161-183.
doi: 10.1007/BF01393691. |
[5] |
L. Escauriaza, Carleman inequalities and the heat operator, Duke Math. J., 104 (2000),
113-127.
doi: 10.1215/S0012-7094-00-10415-2. |
[6] |
L. Escauriaza and F. J. Fernández, Unique continuation for parabolic operators, Ark. Mat., 41 (2003), 35-60.
doi: 10.1007/BF02384566. |
[7] |
L. Escauriaza, F. J. Fernández and S. Vessella, Doubling properties of caloric functions, Appl. Anal., 85 (2006), 205-223.
doi: 10.1080/00036810500277082. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[14] |
F. Lin, A uniqueness theorem for parabolic equations, Commun. Pure Appl. Math., 43 (1990),
127-136.
doi: 10.1002/cpa.3160430105. |
[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. |
[16] |
C. Poon, Unique continuation for parabolic equations, Comm. PDE, 21 (1996), 521-539.
doi: 10.1080/03605309608821195. |
[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. |
[18] |
C. Sogge, A unique continuation theorem for second order parabolic differential operators, Ark. Mat., 28 (1990), 159-182.
doi: 10.1007/BF02387373. |
[19] |
M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Prob., 25 (2009), 123013.
doi: 10.1088/0266-5611/25/12/123013. |
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. |
[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. |
[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. |
[4] |
H. Donnelly and C. Fefferman, Nodal sets of eigenfunctions on Riemann manifolds, Invent. Math., 93 (1988), 161-183.
doi: 10.1007/BF01393691. |
[5] |
L. Escauriaza, Carleman inequalities and the heat operator, Duke Math. J., 104 (2000),
113-127.
doi: 10.1215/S0012-7094-00-10415-2. |
[6] |
L. Escauriaza and F. J. Fernández, Unique continuation for parabolic operators, Ark. Mat., 41 (2003), 35-60.
doi: 10.1007/BF02384566. |
[7] |
L. Escauriaza, F. J. Fernández and S. Vessella, Doubling properties of caloric functions, Appl. Anal., 85 (2006), 205-223.
doi: 10.1080/00036810500277082. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[14] |
F. Lin, A uniqueness theorem for parabolic equations, Commun. Pure Appl. Math., 43 (1990),
127-136.
doi: 10.1002/cpa.3160430105. |
[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. |
[16] |
C. Poon, Unique continuation for parabolic equations, Comm. PDE, 21 (1996), 521-539.
doi: 10.1080/03605309608821195. |
[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. |
[18] |
C. Sogge, A unique continuation theorem for second order parabolic differential operators, Ark. Mat., 28 (1990), 159-182.
doi: 10.1007/BF02387373. |
[19] |
M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Prob., 25 (2009), 123013.
doi: 10.1088/0266-5611/25/12/123013. |
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