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June  2022, 11(3): 711-727. doi: 10.3934/eect.2021022

On the stabilization for the high-order Kadomtsev-Petviashvili and the Zakharov-Kuznetsov equations with localized damping

1. 

Universidade Federal do Piauí, Campus Universitário Ministro Petrônio Portella, Ininga, 64049-550, Teresina, Piauí, Brazil

2. 

Universidade Federal do Ceará, Rua Coronel Estanislau Frota, 563 - Bloco I, Centro - Campus de Sobral, Mucambinho, 62010-560, Sobral, Ceará, Brazil

* Corresponding author: ailtoncnascimento@gmail.com

Received  August 2020 Revised  February 2021 Published  June 2022 Early access  April 2021

In this paper we prove the exponential decay of the energy for the high-order Kadomtsev-Petviashvili II equation with localized damping. To do that, we use the classical dissipation-observability method and a unique continuation principle introduced by Bourgain in [3] here extended for the high-order Kadomtsev-Petviashvili. A similar result is also obtained for the two-dimensional Zakharov-Kuznetsov (ZK)equation. The method of proof works better for the ZK equation, so we were led to make some subtle modifications on it to include KP type equations. In fact, to reach a key estimate we use an anisotropic Gagliardo-Nirenberg inequality to drop the $ y $-derivative of the norm.

Citation: Roger P. de Moura, Ailton C. Nascimento, Gleison N. Santos. On the stabilization for the high-order Kadomtsev-Petviashvili and the Zakharov-Kuznetsov equations with localized damping. Evolution Equations and Control Theory, 2022, 11 (3) : 711-727. doi: 10.3934/eect.2021022
References:
[1]

O. V. Besov, V. P. Il'in and S. M. Nikol'skii, Integral Representations of Functions and Imbedding Theorems, Vol. I., New York-Toronto, Ont.-London, 1978.

[2]

J. Bourgain, On the Cauchy problem for the Kadomtsev-Petviashvili equation, Geom. Funct. Anal., 3 (1993), 315-341.  doi: 10.1007/BF01896259.

[3]

J. Bourgain, On the compactness of the support of solutions of dispersive equations, Int. Math. Res. Notices, 9 (1997), 437-447.  doi: 10.1155/S1073792897000305.

[4]

M. Castelli and G. Doronin, Modified and subcritical Zakharov-Kuznetsov equations posed on rectangles, J. Differential Equations, 275 (2021), 554-580.  doi: 10.1016/j.jde.2020.11.025.

[5]

W. ChenJ. Li and C. Miao, On the low regularity of the fifth order Kadomtsev-Petviashvili I equation, J. Differential Equations, 245 (2008), 3433-3469.  doi: 10.1016/j.jde.2008.07.005.

[6]

G. G. Doronin and N. A. Larkin, Stabilization for the linear Zakharov-Kuznetsov equation without critical size restrictions, J. Math. Anal. Appl., 428 (2015), 337-355.  doi: 10.1016/j.jmaa.2015.03.010.

[7]

G. G. Doronin and N. A. Larkin, Stabilization of regular solutions for the Zakharov-Kuznetsov equation posed on bounded rectangles and on a strip, Proc. Edinb. Math. Soc., 58 (2015), 661-682.  doi: 10.1017/S0013091514000248.

[8]

A. Esfahani and A. Pastor, On the unique continuation property for Kadomtsev-Petviashvili-I and Benjamin-Ono-Zakharov-Kuznetsov equations, Bull. Lond. Math. Soc., 43 (2011), 1130-1140.  doi: 10.1112/blms/bdr048.

[9]

A. V. Faminskii, The Cauchy problem for the Zakharov-Kuznetsov equation, Differ. Equ., 31 (1995), 1002-1012. 

[10]

A. Friedman, Partial Differential Equations, Dover ed., New York, 2008.

[11]

D. A. Gomes and M. Panthee, Exponential energy decay for the Kadomtsev-Petviashvili (KP-II) equation, São Paulo J. Math. Sci., 5 (2011), 135-148.  doi: 10.11606/issn.2316-9028.v5i2p135-148.

[12]

A. Grünrock and S. Herr, The Fourier restriction norm method for the Zakharov-Kuznetsov equation, Discrete Contin. Dyn. Syst., 34 (2014), 2061-2068.  doi: 10.3934/dcds.2014.34.2061.

[13]

A. GrünrockM. Panthee and J. D. Silva, On KP-II type equations on cylinders, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2335-2358.  doi: 10.1016/j.anihpc.2009.04.002.

[14]

M. HadacS. Herr and H. Koch, Well-posedness and scattering for the KP II equation in a critical space, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 917-941.  doi: 10.1016/j.anihpc.2008.04.002.

[15]

M. Hadac, Well-posedness for the Kadomtsev-Petviashvili II equation and generalisations, Trans. Amer. Math. Soc., 360 (2008), 6555-6572.  doi: 10.1090/S0002-9947-08-04515-7.

[16]

J. K. Hunter and J. Scheurle, Existence of perturbed solitary waves to a model equation for water waves, Physica D, 32 (1988), 253-268.  doi: 10.1016/0167-2789(88)90054-1.

[17]

A. Ill'ichevA. Semenov and A. Yu, Stability of solitary waves in dispersive medias described by a fifth-order evolution equation, Theoret. Comput. Fluid Dynamics, 3 (1992), 307-326.  doi: 10.1007/BF00417931.

[18]

A. D. IonescuC. Kenig and D. Tataru, Global well-posedness of the KP-I initial-value problem in the energy space, Invent. Math., 173 (2008), 265-304.  doi: 10.1007/s00222-008-0115-0.

[19]

R. Iório Jr. and W. Nunes, On equations of KP-type, Proc. Roy. Soc. Edinburgh, 128 (1998), 725-743.  doi: 10.1017/S0308210500021740.

[20]

P. IsazaJ. López and J. Mejía, Cauchy problem for the fifth order Kadomtsev-Petviashvili (KPII) equation, Comm. Pure Appl. Anal., 5 (2006), 887-905.  doi: 10.3934/cpaa.2006.5.887.

[21]

P. Isaza and J. Mejía, Global solution for the Kadomtsev-Petviashvili equation (KPII) in anisotropic Sobolev spaces of negative indices, Electron. J. Differential Equations, 68 (2003), 1-12. 

[22]

P. Isaza, F. Linares and G. Ponce, On the propagation of regularity of solutions of the Kadomtsev-Petviashvili equation, SIAM J. Math. Anal., textbf48 (2016), 1006-1024. doi: 10.1137/15M1012098.

[23]

B. B. Kadomtsev and V. I. Petviashvili, On the stability of solitary waves in weakly dispersing media, Sov. Phys. Dokl., 15 (1970), 539-541. 

[24]

T. Kawahara, Oscillatory solitary waves in dispersive media, J. Phys. Soc. Japan, 33 (1972), 260-264.  doi: 10.1143/JPSJ.33.260.

[25]

S. Kinoshita, Global Well-posedness for the Cauchy problem of the Zakharov-Kuznetsov equation in 2D, Ann. Inst. H. Poincaré Anal. Non Linéaire, 38 (2021), 451-505.  doi: 10.1016/j.anihpc.2020.08.003.

[26]

C. Klein and J.-C. Saut, Numerical study of blow up and stability of solutions of generalized Kadomtsev-Petviashvili equations, J. Nonlinear Sci., 22 (2012), 763-811.  doi: 10.1007/s00332-012-9127-4.

[27]

D. LannesF. Linares and J.-C. Saut, The Cauchy problem for the Euler-Poisson system and derivation of the Zakharov-Kuznetsov equation, Studies in Phase Space Analysis with Applications to PDEs, Progr. Nonlinear Differential Equations Appl., 84 (2013), 181-213.  doi: 10.1007/978-1-4614-6348-1_10.

[28]

N. A. Larkin, Decay of regular solutions for the critical 2D Zakharov-Kuznetsov equation posed on rectangles, J. Math. Phys., 61 (2020), 061509, 1-13. doi: 10.1063/1.5100284.

[29]

F. Linares and A. Pazoto, On the exponential decay of the critical generalized Korteweg-de Vries equation with localized damping, Proc. Amer. Math. Soc., 135 (2007), 1515-1522.  doi: 10.1090/S0002-9939-07-08810-7.

[30]

F. Linares and A. Pastor, Well-posedness for the two-dimensional modified Zakharov-Kuznetsov equation, SIAM J. Math. Anal., 41 (2009), 1323-1339.  doi: 10.1137/080739173.

[31]

J. Li and S. Shi, A local well-posed result for the fifth order KP-II initial value problem, J. Math. Anal. Appl., 402 (2013), 679-692.  doi: 10.1016/j.jmaa.2013.01.069.

[32]

J. Li and J. Xiao, Well-posedness of the fifth order Kadomtsev-Petviashvili I equation in anisotropic Sobolev spaces with nonnegative indices, J. Math. Pures Appl., 90 (2008), 338-352.  doi: 10.1016/j.matpur.2008.06.005.

[33]

J. Li and X. Li, Well-posedness for the fifth order KP-II initial data problem in $H^{s, 0}(\mathbb R\times \mathbb T)$, J. Differential Equations, 262 (2017), 2196-2230.  doi: 10.1016/j.jde.2016.10.048.

[34]

L. Molinet and D. Pilod, Bilinear Strichartz estimates for the Zakharov-Kuznetsov equation and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 347-371.  doi: 10.1016/j.anihpc.2013.12.003.

[35]

A. C. Nascimento, On the propagation of regularities in solutions of the fifth order Kadomtsev-Petviashvili II equation, J. Math. Anal. App., 478 (2019), 156-181.  doi: 10.1016/j.jmaa.2019.05.024.

[36]

A. Pazy, Semigroups os Linear Operators and Applications to Partial Differential Equations, Springer Verlag, New York, USA, 1983.

[37]

M. Panthee, A note on the unique continuation property for Zakharov-Kuznetsov equation, Nonlin. Ann., 59 (2004), 425-438.  doi: 10.1016/j.na.2004.07.022.

[38]

M. Panthee, Unique continuation property for the Kadomtsev-Petviashvili (KP-II) equation, Electron. J. Differential Equations, 59 (2005), 1-12. 

[39]

G. Perla MenzalaC. F. Vasconcellos and E. Zuazua, Stabilization of the Korteweg-de Vries equation with localized damping, Quart. Appl. Math., 60 (2002), 111-129.  doi: 10.1090/qam/1878262.

[40]

L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESAIN, Controll Optimization Calculus of Variations, 2 (1997), 33-55.  doi: 10.1051/cocv:1997102.

[41]

L. Rosier and B. Y. Zhang, Global stabilization of the generalized Korteweg-de Vries equation posed on a finite domain, SIAM J. Contr. Opt., 45 (2006), 927-956.  doi: 10.1137/050631409.

[42]

J. -C. Saut and N. Tzvetkov, The Cauchy problem for higher-order KP equations, J. Differential Equations, 153 (1999), 196-222.  doi: 10.1006/jdeq.1998.3534.

[43]

J. -C. Saut and N. Tzvetkov, The Cauchy problem for the fifth order KP equations, J. Math. Pures Appl. (9), 79 (2000), 307-338.  doi: 10.1016/S0021-7824(00)00156-2.

[44]

C. F. Vasconcellos and P. N. da Silva, Stabilization of the Kawahara equation with localized damping, ESAIM Control Optim. Calc. Var., 17 (2011), 102-116.  doi: 10.1051/cocv/2009041.

[45]

V. Zakharov and E. Kuznetsov, Three-dimensional solitons, Sov. Phys. JETP. 39 (1974), 285-286.

[46]

E. Zuazua, Controllability of Partial Differential Equations, 3rd cycle. Castro Urdiales (Es- pagne), (2006), 1-311.

[47]

E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Comm. Partial Differential Equations, 15 (1990), 205-235.  doi: 10.1080/03605309908820684.

show all references

References:
[1]

O. V. Besov, V. P. Il'in and S. M. Nikol'skii, Integral Representations of Functions and Imbedding Theorems, Vol. I., New York-Toronto, Ont.-London, 1978.

[2]

J. Bourgain, On the Cauchy problem for the Kadomtsev-Petviashvili equation, Geom. Funct. Anal., 3 (1993), 315-341.  doi: 10.1007/BF01896259.

[3]

J. Bourgain, On the compactness of the support of solutions of dispersive equations, Int. Math. Res. Notices, 9 (1997), 437-447.  doi: 10.1155/S1073792897000305.

[4]

M. Castelli and G. Doronin, Modified and subcritical Zakharov-Kuznetsov equations posed on rectangles, J. Differential Equations, 275 (2021), 554-580.  doi: 10.1016/j.jde.2020.11.025.

[5]

W. ChenJ. Li and C. Miao, On the low regularity of the fifth order Kadomtsev-Petviashvili I equation, J. Differential Equations, 245 (2008), 3433-3469.  doi: 10.1016/j.jde.2008.07.005.

[6]

G. G. Doronin and N. A. Larkin, Stabilization for the linear Zakharov-Kuznetsov equation without critical size restrictions, J. Math. Anal. Appl., 428 (2015), 337-355.  doi: 10.1016/j.jmaa.2015.03.010.

[7]

G. G. Doronin and N. A. Larkin, Stabilization of regular solutions for the Zakharov-Kuznetsov equation posed on bounded rectangles and on a strip, Proc. Edinb. Math. Soc., 58 (2015), 661-682.  doi: 10.1017/S0013091514000248.

[8]

A. Esfahani and A. Pastor, On the unique continuation property for Kadomtsev-Petviashvili-I and Benjamin-Ono-Zakharov-Kuznetsov equations, Bull. Lond. Math. Soc., 43 (2011), 1130-1140.  doi: 10.1112/blms/bdr048.

[9]

A. V. Faminskii, The Cauchy problem for the Zakharov-Kuznetsov equation, Differ. Equ., 31 (1995), 1002-1012. 

[10]

A. Friedman, Partial Differential Equations, Dover ed., New York, 2008.

[11]

D. A. Gomes and M. Panthee, Exponential energy decay for the Kadomtsev-Petviashvili (KP-II) equation, São Paulo J. Math. Sci., 5 (2011), 135-148.  doi: 10.11606/issn.2316-9028.v5i2p135-148.

[12]

A. Grünrock and S. Herr, The Fourier restriction norm method for the Zakharov-Kuznetsov equation, Discrete Contin. Dyn. Syst., 34 (2014), 2061-2068.  doi: 10.3934/dcds.2014.34.2061.

[13]

A. GrünrockM. Panthee and J. D. Silva, On KP-II type equations on cylinders, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 2335-2358.  doi: 10.1016/j.anihpc.2009.04.002.

[14]

M. HadacS. Herr and H. Koch, Well-posedness and scattering for the KP II equation in a critical space, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 917-941.  doi: 10.1016/j.anihpc.2008.04.002.

[15]

M. Hadac, Well-posedness for the Kadomtsev-Petviashvili II equation and generalisations, Trans. Amer. Math. Soc., 360 (2008), 6555-6572.  doi: 10.1090/S0002-9947-08-04515-7.

[16]

J. K. Hunter and J. Scheurle, Existence of perturbed solitary waves to a model equation for water waves, Physica D, 32 (1988), 253-268.  doi: 10.1016/0167-2789(88)90054-1.

[17]

A. Ill'ichevA. Semenov and A. Yu, Stability of solitary waves in dispersive medias described by a fifth-order evolution equation, Theoret. Comput. Fluid Dynamics, 3 (1992), 307-326.  doi: 10.1007/BF00417931.

[18]

A. D. IonescuC. Kenig and D. Tataru, Global well-posedness of the KP-I initial-value problem in the energy space, Invent. Math., 173 (2008), 265-304.  doi: 10.1007/s00222-008-0115-0.

[19]

R. Iório Jr. and W. Nunes, On equations of KP-type, Proc. Roy. Soc. Edinburgh, 128 (1998), 725-743.  doi: 10.1017/S0308210500021740.

[20]

P. IsazaJ. López and J. Mejía, Cauchy problem for the fifth order Kadomtsev-Petviashvili (KPII) equation, Comm. Pure Appl. Anal., 5 (2006), 887-905.  doi: 10.3934/cpaa.2006.5.887.

[21]

P. Isaza and J. Mejía, Global solution for the Kadomtsev-Petviashvili equation (KPII) in anisotropic Sobolev spaces of negative indices, Electron. J. Differential Equations, 68 (2003), 1-12. 

[22]

P. Isaza, F. Linares and G. Ponce, On the propagation of regularity of solutions of the Kadomtsev-Petviashvili equation, SIAM J. Math. Anal., textbf48 (2016), 1006-1024. doi: 10.1137/15M1012098.

[23]

B. B. Kadomtsev and V. I. Petviashvili, On the stability of solitary waves in weakly dispersing media, Sov. Phys. Dokl., 15 (1970), 539-541. 

[24]

T. Kawahara, Oscillatory solitary waves in dispersive media, J. Phys. Soc. Japan, 33 (1972), 260-264.  doi: 10.1143/JPSJ.33.260.

[25]

S. Kinoshita, Global Well-posedness for the Cauchy problem of the Zakharov-Kuznetsov equation in 2D, Ann. Inst. H. Poincaré Anal. Non Linéaire, 38 (2021), 451-505.  doi: 10.1016/j.anihpc.2020.08.003.

[26]

C. Klein and J.-C. Saut, Numerical study of blow up and stability of solutions of generalized Kadomtsev-Petviashvili equations, J. Nonlinear Sci., 22 (2012), 763-811.  doi: 10.1007/s00332-012-9127-4.

[27]

D. LannesF. Linares and J.-C. Saut, The Cauchy problem for the Euler-Poisson system and derivation of the Zakharov-Kuznetsov equation, Studies in Phase Space Analysis with Applications to PDEs, Progr. Nonlinear Differential Equations Appl., 84 (2013), 181-213.  doi: 10.1007/978-1-4614-6348-1_10.

[28]

N. A. Larkin, Decay of regular solutions for the critical 2D Zakharov-Kuznetsov equation posed on rectangles, J. Math. Phys., 61 (2020), 061509, 1-13. doi: 10.1063/1.5100284.

[29]

F. Linares and A. Pazoto, On the exponential decay of the critical generalized Korteweg-de Vries equation with localized damping, Proc. Amer. Math. Soc., 135 (2007), 1515-1522.  doi: 10.1090/S0002-9939-07-08810-7.

[30]

F. Linares and A. Pastor, Well-posedness for the two-dimensional modified Zakharov-Kuznetsov equation, SIAM J. Math. Anal., 41 (2009), 1323-1339.  doi: 10.1137/080739173.

[31]

J. Li and S. Shi, A local well-posed result for the fifth order KP-II initial value problem, J. Math. Anal. Appl., 402 (2013), 679-692.  doi: 10.1016/j.jmaa.2013.01.069.

[32]

J. Li and J. Xiao, Well-posedness of the fifth order Kadomtsev-Petviashvili I equation in anisotropic Sobolev spaces with nonnegative indices, J. Math. Pures Appl., 90 (2008), 338-352.  doi: 10.1016/j.matpur.2008.06.005.

[33]

J. Li and X. Li, Well-posedness for the fifth order KP-II initial data problem in $H^{s, 0}(\mathbb R\times \mathbb T)$, J. Differential Equations, 262 (2017), 2196-2230.  doi: 10.1016/j.jde.2016.10.048.

[34]

L. Molinet and D. Pilod, Bilinear Strichartz estimates for the Zakharov-Kuznetsov equation and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 347-371.  doi: 10.1016/j.anihpc.2013.12.003.

[35]

A. C. Nascimento, On the propagation of regularities in solutions of the fifth order Kadomtsev-Petviashvili II equation, J. Math. Anal. App., 478 (2019), 156-181.  doi: 10.1016/j.jmaa.2019.05.024.

[36]

A. Pazy, Semigroups os Linear Operators and Applications to Partial Differential Equations, Springer Verlag, New York, USA, 1983.

[37]

M. Panthee, A note on the unique continuation property for Zakharov-Kuznetsov equation, Nonlin. Ann., 59 (2004), 425-438.  doi: 10.1016/j.na.2004.07.022.

[38]

M. Panthee, Unique continuation property for the Kadomtsev-Petviashvili (KP-II) equation, Electron. J. Differential Equations, 59 (2005), 1-12. 

[39]

G. Perla MenzalaC. F. Vasconcellos and E. Zuazua, Stabilization of the Korteweg-de Vries equation with localized damping, Quart. Appl. Math., 60 (2002), 111-129.  doi: 10.1090/qam/1878262.

[40]

L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESAIN, Controll Optimization Calculus of Variations, 2 (1997), 33-55.  doi: 10.1051/cocv:1997102.

[41]

L. Rosier and B. Y. Zhang, Global stabilization of the generalized Korteweg-de Vries equation posed on a finite domain, SIAM J. Contr. Opt., 45 (2006), 927-956.  doi: 10.1137/050631409.

[42]

J. -C. Saut and N. Tzvetkov, The Cauchy problem for higher-order KP equations, J. Differential Equations, 153 (1999), 196-222.  doi: 10.1006/jdeq.1998.3534.

[43]

J. -C. Saut and N. Tzvetkov, The Cauchy problem for the fifth order KP equations, J. Math. Pures Appl. (9), 79 (2000), 307-338.  doi: 10.1016/S0021-7824(00)00156-2.

[44]

C. F. Vasconcellos and P. N. da Silva, Stabilization of the Kawahara equation with localized damping, ESAIM Control Optim. Calc. Var., 17 (2011), 102-116.  doi: 10.1051/cocv/2009041.

[45]

V. Zakharov and E. Kuznetsov, Three-dimensional solitons, Sov. Phys. JETP. 39 (1974), 285-286.

[46]

E. Zuazua, Controllability of Partial Differential Equations, 3rd cycle. Castro Urdiales (Es- pagne), (2006), 1-311.

[47]

E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Comm. Partial Differential Equations, 15 (1990), 205-235.  doi: 10.1080/03605309908820684.

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