Global Carleman estimate and its applications for a sixth-order equation related to thin solid films

Previous Article
Sharp subcritical Sobolev inequalities and uniqueness of nonnegative solutions to high-order Lane-Emden equations on $ \mathbb{S}^n $
CPAA Home
This Issue
Next Article
On spectral and fractional powers of damped wave equations

August 2022, 21(8): 2775-2797. doi: 10.3934/cpaa.2022072

Global Carleman estimate and its applications for a sixth-order equation related to thin solid films

Yanpeng Jin and Ying Fu ^,

Center for nonlinear studies and school of Mathematics, Northwest University, Xi'an 710127, China

^* Corresponding author

Received December 2021 Published August 2022 Early access April 2022

Fund Project: The second author is supported by NSF grant-11471259 and 11631007 and the National Science Basic Research Program of Shaanxi Province (Program No. 2019JM-007 and 2020JC-37)

Considered herein is the initial boundary value problem associated with a sixth-order nonlinear parabolic equation in a bounded domain. We first establish a new global Carleman estimate for the sixth-order parabolic operator. Based on this estimate, we obtain the local exact controllability to the trajectories and the unique continuation property of the parabolic equation.

Keywords: Carleman estimates, local exact controllability, unique continuation property.

Mathematics Subject Classification: Primary: 35K25, 93B05; Secondary: 35B60, 93C20.

Citation: 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, 2022, 21 (8) : 2775-2797. doi: 10.3934/cpaa.2022072

References:

[1]	V. M. Alekseev, V. M. Tikhomirov and S. V. Fomin, Optimal Control, Contemporary Soviet Mathematics, Consultants Bureau, New York, 1987. doi: 10.1007/978-1-4615-7551-1.
[2]	E. Bustamante, J. J. Urrea and J. Mejía, On the unique continuation property of solutions of the three-dimensional Zakharov-Kuznetsov equation, Nonlinear Anal. Real World Appl., 39 (2018), 537-553. doi: 10.1016/j.nonrwa.2017.08.003.
[3]	T. Carleman, Sur un problème d'unicit$\acute{e}$ pur les syst$\grave{e}$mes d'$\acute{e}$quations aux d$\acute{e}$riv$\acute{e}$es partielles $\grave{a}$ deux variables ind$\acute{e}$pendantes, Ark. Mat. Astr. Fys., 26 (1939), 9pp.
[4]	E. Cerpa and A. Mercado, Local exact controllability to the trajectories of the 1-D Kuramoto-Sivashinsky equation, J. Differ. Equ., 250 (2011), 2024-2044. doi: 10.1016/j.jde.2010.12.015.
[5]	E. Cerpa, C. Montoya and B.Y. Zhang, Local exact controllability to the trajectories of the Korteweg-de Vries-Burgers equation on a bounded domain with mixed boundary conditions, J. Differ. Equ., 268 (2020), 4975-4972. doi: 10.1016/j.jde.2019.10.043.
[6]	M. Chen, Unique continuation property for the Zakharov-Kuznetsov equation, Comput. Math. Appl., 77 (2019), 1273-1281. doi: 10.1016/j.camwa.2018.11.002.
[7]	M. Davila and G. P. Menzala, Unique continuation for the Benjamin-Bona-Mahony and Boussinesq's equations, NoDEA Nonlinear Differ. Equ. Appl., 5 (1998), 367-382. doi: 10.1007/s000300050051.
[8]	P. Gao, A new global Carleman estimate for the one-dimensional Kuramoto-Sivashinsky equation and applications to exact controllability to the trajectories and an inverse problem, Nonlinear Anal., 117 (2015), 133-147. doi: 10.1016/j.na.2015.01.015.
[9]	P. Gao, Local exact controllability to the trajectories of the Swift-Hohenberg equation, Nonlinear Anal., 139 (2016), 169-195. doi: 10.1016/j.na.2016.02.023.
[10]	P. Gao, Carleman estimates and unique continuation property for 1-D viscous Camassa-Holm equation, Discrete Contin. Dyn. Syst., 37 (2017), 169-188. doi: 10.3934/dcds.2017007.
[11]	P. Gao, Global Carleman estimate for the Kawahara equation and its applications, Commun. Pure Appl. Anal., 17 (2018), 1853-1874. doi: 10.3934/cpaa.2018088.
[12]	P. Gao, Global exact controllability to the trajectories of the Kuramoto-Sivashinsky equation, Evol. Equ. Control Theory, 9 (2020), 181-191. doi: 10.3934/eect.2020002.
[13]	O. Glass and S. Guerrero, Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit, Asymptot. Anal., 60 (2008), 61-100.
[14]	O. Glass and S. Guerrero, On the controllability of the fifth-order Korteweg-de Vries equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2181-2209. doi: 10.1016/j.anihpc.2009.01.010.
[15]	A. A. Golovin, S. H. Davis and P. W. Voorhees, Self-organization of quantum dots in epitaxially strained solid films, Phys. Rev. E, 68 (2003), 056203, 11pp.
[16]	S. Guerrero and K. Kassab, Carleman estimate and null controllability of a fourth order parabolic equation in dimension $ N\geq 2$, J. Math. Pures Appl., 121 (2019), 135-161. doi: 10.1016/j.matpur.2018.04.004.
[17]	S. Guerrero and C. Montoya, Local null controllability of the N-dimensional Navier-Stokes system with nonlinear Navier-slip boundary conditions and N-1 scalar controls, J. Math. Pures Appl., 113 (2018), 37-69. doi: 10.1016/j.matpur.2018.03.004.
[18]	P. Guzm$\acute{a}$n, Local exact controllability to the trajectories of the Cahn-Hilliard equation, Appl. Math. Optim., 82 (2020), 279-306. doi: 10.1007/s00245-018-9500-2.
[19]	O. Y. Imanuvilov and M. Yamamoto, Carleman estimate for a parabolic equation in a Sobolev space of negative order and its applications, Lecture Notes Pure Appl. Math., 218 (2001), 113-137.
[20]	C. E. Kenig, G. Ponce and L. Vega, On unique continuation for nonlinear Schr$\ddot{o}$dinger equations, Commun. Pure Appl. Math., 56 (2003), 1247-1262. doi: 10.1002/cpa.10094.
[21]	X. C. Liu and C. Z. Qu, Existence and blow-up of weak solutions for a sixth-order equation related to thin solid films, Nonlinear Anal. Real World Appl., 11 (2010), 4214-4222. doi: 10.1016/j.nonrwa.2010.05.008.
[22]	D. Mitra, Local null controllability of viscous Camassa-Holm equation, J. Evol. Equ., 18 (2018), 627-657. doi: 10.1007/s00028-017-0414-2.
[23]	M. Panthee, A note on the unique continuation property for Zakharov-Kuznetsov equation, Nonlinear Anal., 59 (2004), 425-438. doi: 10.1016/j.na.2004.07.022.
[24]	L. Rosier and B. Y. Zhang, Global stabilization of the generalized Korteweg-de Vries equation posed on a finite domain, SIAM J. Control Optim., 45 (2006), 927-956. doi: 10.1137/050631409.
[25]	L. Rosier and B. Y. Zhang, Unique continuation property and control for the Benjamin-Bona-Mahony equation on a periodic domain, J. Differ. Equ., 254 (2013), 141-178. doi: 10.1016/j.jde.2012.08.014.
[26]	P. N. da Silva, Unique continuation for the Kawahara equation, TEMA Tend. Mat. Apl. Comput., 8 (2007), 463-473. doi: 10.5540/tema.2007.08.03.0463.
[27]	B. Y. Zhang, Unique continuation for the Korteweg-de Vries equation, SIAM J. Math. Anal., 23 (1992), 55-71. doi: 10.1137/0523004.
[28]	B. Y. Zhang, Unique continuation properties of the nonlinear Schr$\ddot{o}$dinger equation, Proc. Roy. Soc. Edinb. Sect. A, 127 (1997), 191-205. doi: 10.1017/S0308210500023581.
[29]	X. Zhang and E. Zuazua, Unique continuation for the linearized Benjamin-Bona-Mahony equation with space-dependent potential, Math. Ann., 325 (2003), 543-582. doi: 10.1007/s00208-002-0391-8.

show all references

References:

[1]	V. M. Alekseev, V. M. Tikhomirov and S. V. Fomin, Optimal Control, Contemporary Soviet Mathematics, Consultants Bureau, New York, 1987. doi: 10.1007/978-1-4615-7551-1.
[2]	E. Bustamante, J. J. Urrea and J. Mejía, On the unique continuation property of solutions of the three-dimensional Zakharov-Kuznetsov equation, Nonlinear Anal. Real World Appl., 39 (2018), 537-553. doi: 10.1016/j.nonrwa.2017.08.003.
[3]	T. Carleman, Sur un problème d'unicit$\acute{e}$ pur les syst$\grave{e}$mes d'$\acute{e}$quations aux d$\acute{e}$riv$\acute{e}$es partielles $\grave{a}$ deux variables ind$\acute{e}$pendantes, Ark. Mat. Astr. Fys., 26 (1939), 9pp.
[4]	E. Cerpa and A. Mercado, Local exact controllability to the trajectories of the 1-D Kuramoto-Sivashinsky equation, J. Differ. Equ., 250 (2011), 2024-2044. doi: 10.1016/j.jde.2010.12.015.
[5]	E. Cerpa, C. Montoya and B.Y. Zhang, Local exact controllability to the trajectories of the Korteweg-de Vries-Burgers equation on a bounded domain with mixed boundary conditions, J. Differ. Equ., 268 (2020), 4975-4972. doi: 10.1016/j.jde.2019.10.043.
[6]	M. Chen, Unique continuation property for the Zakharov-Kuznetsov equation, Comput. Math. Appl., 77 (2019), 1273-1281. doi: 10.1016/j.camwa.2018.11.002.
[7]	M. Davila and G. P. Menzala, Unique continuation for the Benjamin-Bona-Mahony and Boussinesq's equations, NoDEA Nonlinear Differ. Equ. Appl., 5 (1998), 367-382. doi: 10.1007/s000300050051.
[8]	P. Gao, A new global Carleman estimate for the one-dimensional Kuramoto-Sivashinsky equation and applications to exact controllability to the trajectories and an inverse problem, Nonlinear Anal., 117 (2015), 133-147. doi: 10.1016/j.na.2015.01.015.
[9]	P. Gao, Local exact controllability to the trajectories of the Swift-Hohenberg equation, Nonlinear Anal., 139 (2016), 169-195. doi: 10.1016/j.na.2016.02.023.
[10]	P. Gao, Carleman estimates and unique continuation property for 1-D viscous Camassa-Holm equation, Discrete Contin. Dyn. Syst., 37 (2017), 169-188. doi: 10.3934/dcds.2017007.
[11]	P. Gao, Global Carleman estimate for the Kawahara equation and its applications, Commun. Pure Appl. Anal., 17 (2018), 1853-1874. doi: 10.3934/cpaa.2018088.
[12]	P. Gao, Global exact controllability to the trajectories of the Kuramoto-Sivashinsky equation, Evol. Equ. Control Theory, 9 (2020), 181-191. doi: 10.3934/eect.2020002.
[13]	O. Glass and S. Guerrero, Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit, Asymptot. Anal., 60 (2008), 61-100.
[14]	O. Glass and S. Guerrero, On the controllability of the fifth-order Korteweg-de Vries equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2181-2209. doi: 10.1016/j.anihpc.2009.01.010.
[15]	A. A. Golovin, S. H. Davis and P. W. Voorhees, Self-organization of quantum dots in epitaxially strained solid films, Phys. Rev. E, 68 (2003), 056203, 11pp.
[16]	S. Guerrero and K. Kassab, Carleman estimate and null controllability of a fourth order parabolic equation in dimension $ N\geq 2$, J. Math. Pures Appl., 121 (2019), 135-161. doi: 10.1016/j.matpur.2018.04.004.
[17]	S. Guerrero and C. Montoya, Local null controllability of the N-dimensional Navier-Stokes system with nonlinear Navier-slip boundary conditions and N-1 scalar controls, J. Math. Pures Appl., 113 (2018), 37-69. doi: 10.1016/j.matpur.2018.03.004.
[18]	P. Guzm$\acute{a}$n, Local exact controllability to the trajectories of the Cahn-Hilliard equation, Appl. Math. Optim., 82 (2020), 279-306. doi: 10.1007/s00245-018-9500-2.
[19]	O. Y. Imanuvilov and M. Yamamoto, Carleman estimate for a parabolic equation in a Sobolev space of negative order and its applications, Lecture Notes Pure Appl. Math., 218 (2001), 113-137.
[20]	C. E. Kenig, G. Ponce and L. Vega, On unique continuation for nonlinear Schr$\ddot{o}$dinger equations, Commun. Pure Appl. Math., 56 (2003), 1247-1262. doi: 10.1002/cpa.10094.
[21]	X. C. Liu and C. Z. Qu, Existence and blow-up of weak solutions for a sixth-order equation related to thin solid films, Nonlinear Anal. Real World Appl., 11 (2010), 4214-4222. doi: 10.1016/j.nonrwa.2010.05.008.
[22]	D. Mitra, Local null controllability of viscous Camassa-Holm equation, J. Evol. Equ., 18 (2018), 627-657. doi: 10.1007/s00028-017-0414-2.
[23]	M. Panthee, A note on the unique continuation property for Zakharov-Kuznetsov equation, Nonlinear Anal., 59 (2004), 425-438. doi: 10.1016/j.na.2004.07.022.
[24]	L. Rosier and B. Y. Zhang, Global stabilization of the generalized Korteweg-de Vries equation posed on a finite domain, SIAM J. Control Optim., 45 (2006), 927-956. doi: 10.1137/050631409.
[25]	L. Rosier and B. Y. Zhang, Unique continuation property and control for the Benjamin-Bona-Mahony equation on a periodic domain, J. Differ. Equ., 254 (2013), 141-178. doi: 10.1016/j.jde.2012.08.014.
[26]	P. N. da Silva, Unique continuation for the Kawahara equation, TEMA Tend. Mat. Apl. Comput., 8 (2007), 463-473. doi: 10.5540/tema.2007.08.03.0463.
[27]	B. Y. Zhang, Unique continuation for the Korteweg-de Vries equation, SIAM J. Math. Anal., 23 (1992), 55-71. doi: 10.1137/0523004.
[28]	B. Y. Zhang, Unique continuation properties of the nonlinear Schr$\ddot{o}$dinger equation, Proc. Roy. Soc. Edinb. Sect. A, 127 (1997), 191-205. doi: 10.1017/S0308210500023581.
[29]	X. Zhang and E. Zuazua, Unique continuation for the linearized Benjamin-Bona-Mahony equation with space-dependent potential, Math. Ann., 325 (2003), 543-582. doi: 10.1007/s00208-002-0391-8.

[1]	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
[2]	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
[3]	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
[4]	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
[5]	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
[6]	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
[7]	El Mustapha Ait Ben Hassi, Farid Ammar khodja, Abdelkarim Hajjaj, Lahcen Maniar. Carleman Estimates and null controllability of coupled degenerate systems. Evolution Equations and Control Theory, 2013, 2 (3) : 441-459. doi: 10.3934/eect.2013.2.441
[8]	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
[9]	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
[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]	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
[12]	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
[13]	Thuy N. T. Nguyen. Carleman estimates for semi-discrete parabolic operators with a discontinuous diffusion coefficient and applications to controllability. Mathematical Control and Related Fields, 2014, 4 (2) : 203-259. doi: 10.3934/mcrf.2014.4.203
[14]	Genni Fragnelli. Null controllability of degenerate parabolic equations in non divergence form via Carleman estimates. Discrete and Continuous Dynamical Systems - S, 2013, 6 (3) : 687-701. doi: 10.3934/dcdss.2013.6.687
[15]	Bao-Zhu Guo, Liang Zhang. Local exact controllability to positive trajectory for parabolic system of chemotaxis. Mathematical Control and Related Fields, 2016, 6 (1) : 143-165. doi: 10.3934/mcrf.2016.6.143
[16]	Cung The Anh, Vu Manh Toi. Local exact controllability to trajectories of the magneto-micropolar fluid equations. Evolution Equations and Control Theory, 2017, 6 (3) : 357-379. doi: 10.3934/eect.2017019
[17]	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
[18]	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
[19]	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
[20]	Can Zhang. Quantitative unique continuation for the heat equation with Coulomb potentials. Mathematical Control and Related Fields, 2018, 8 (3&4) : 1097-1116. doi: 10.3934/mcrf.2018047

2021 Impact Factor: 1.273

Tools

Metrics

Other articles
by authors

on AIMS
Yanpeng Jin Ying Fu
on Google Scholar
Yanpeng Jin Ying Fu

2021 Impact Factor: 1.273

Tools

Article outline

2021 Impact Factor: 1.273

Tools

Metrics

Other articles
by authors

on AIMS
Yanpeng Jin Ying Fu
on Google Scholar
Yanpeng Jin Ying Fu

[Back to Top]