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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

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.

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. BustamanteJ. 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. CerpaC. 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. KenigG. 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. BustamanteJ. 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. CerpaC. 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. KenigG. 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.

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