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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 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 & Control Theory, 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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[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.  Google Scholar

[38]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[45]

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

[46]

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

[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.  Google Scholar

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[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.  Google Scholar

[38]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[45]

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

[46]

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

[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.  Google Scholar

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