July  2011, 10(4): 1239-1255. doi: 10.3934/cpaa.2011.10.1239

On the support of solutions to the Kadomtsev-Petviashvili (KP-II) equation

1. 

Departamento de Matemáticas, Universidad Nacional de Colombia, A. A. 3840 Medellín, Colombia, Colombia

Received  February 2010 Revised  October 2010 Published  April 2011

In this article we prove that sufficiently smooth solutions of the Kadomtsev-Petviashvili (KP-II) equation:

$ \partial _t u+\partial^3_x u+\partial^{-1}_x\partial^2_y u+u\partial_x u =0, $

that have compact support for two different times are identically zero.

Citation: Pedro Isaza, Jorge Mejía. On the support of solutions to the Kadomtsev-Petviashvili (KP-II) equation. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1239-1255. doi: 10.3934/cpaa.2011.10.1239
References:
[1]

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

[2]

L. Escauriaza, C. Kenig, G. Ponce and L. Vega, On uniqueness properties of solutions of the k-Generalized KdV equations,, J. Funct. Anal., {244 (2007), 504. doi: 10.1016/j.jfa.2006.11.004.

[3]

L. Escauriaza, C. Kenig, G. Ponce and L. Vega, Convexity properties of solutions to the free Schrödinger equation with Gaussian decay,, Math. Res. Lett., {15 (2008), 957.

[4]

A. S. Fokas and L. Y. Sung, On the solvability of the N-wave, Davey-Stewartson and Kadomtsev-Petviashvili equations,, Inverse Problems, {8 (1992), 673. doi: 10.1088/0266-5611/8/5/002.

[5]

M. Hadac, S. Herr and H. Koch, Well-posedness and scattering for the KP-II equation in a critical space,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, {26 (2009), 917. doi: 10.1016/j.anihpc.2008.04.002.

[6]

P. Isaza and J. Mejía, Local and global Cauchy problems for the Kadomtsev-Petviashvili (KP-II) equation in Sobolev spaces of negative indices,, Comm. Partial Differential Equations, {26 (2001), 1027. doi: 10.1081/PDE-100002387.

[7]

P. Isaza and J. Mejía, Global solutions for the Kadomtsev-Petviashvili (KP-II) equation in anisotropic Sobolev spaces of negative indices,, Electron. J. Differential Equations, {2003 (2003), 1.

[8]

P. Isaza and J. Mejía, On the support of solutions to the Ostrovsky equation with negative dispersion,, J. Differential Equations, {247 (2009), 1851. doi: 10.1016/j.jde.2009.03.022.

[9]

P. Isaza and J. Mejía, On the support of solutions to the Ostrovsky equation with positive dispersion,, Nonlinear Anal., {72 (2010), 4016. doi: 10.1016/j.na.2010.01.033.

[10]

C. Kenig, G. Ponce and L. Vega, On the support of solutions to the generalized KdV equation,, Ann. Inst. H. Poincar\'e Anal. Non lin\'eaire, {19 (2002), 191. doi: 10.1016/S0294-1449(01)00073-7.

[11]

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

[12]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, (1983).

[13]

J. C. Saut and B. Scheurer, Unique continuation for some evolution equations,, J. Differential Equations, {66 (1987), 118. doi: 10.1016/0022-0396(87)90043-X.

[14]

H. Takaoka and N. Tzvetkov, On the local regularity of Kadomtsev-Petviashvili-II equation,, Internat. Math. Res. Notices, 2 (2001), 77. doi: 10.1155/S1073792801000058.

show all references

References:
[1]

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

[2]

L. Escauriaza, C. Kenig, G. Ponce and L. Vega, On uniqueness properties of solutions of the k-Generalized KdV equations,, J. Funct. Anal., {244 (2007), 504. doi: 10.1016/j.jfa.2006.11.004.

[3]

L. Escauriaza, C. Kenig, G. Ponce and L. Vega, Convexity properties of solutions to the free Schrödinger equation with Gaussian decay,, Math. Res. Lett., {15 (2008), 957.

[4]

A. S. Fokas and L. Y. Sung, On the solvability of the N-wave, Davey-Stewartson and Kadomtsev-Petviashvili equations,, Inverse Problems, {8 (1992), 673. doi: 10.1088/0266-5611/8/5/002.

[5]

M. Hadac, S. Herr and H. Koch, Well-posedness and scattering for the KP-II equation in a critical space,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, {26 (2009), 917. doi: 10.1016/j.anihpc.2008.04.002.

[6]

P. Isaza and J. Mejía, Local and global Cauchy problems for the Kadomtsev-Petviashvili (KP-II) equation in Sobolev spaces of negative indices,, Comm. Partial Differential Equations, {26 (2001), 1027. doi: 10.1081/PDE-100002387.

[7]

P. Isaza and J. Mejía, Global solutions for the Kadomtsev-Petviashvili (KP-II) equation in anisotropic Sobolev spaces of negative indices,, Electron. J. Differential Equations, {2003 (2003), 1.

[8]

P. Isaza and J. Mejía, On the support of solutions to the Ostrovsky equation with negative dispersion,, J. Differential Equations, {247 (2009), 1851. doi: 10.1016/j.jde.2009.03.022.

[9]

P. Isaza and J. Mejía, On the support of solutions to the Ostrovsky equation with positive dispersion,, Nonlinear Anal., {72 (2010), 4016. doi: 10.1016/j.na.2010.01.033.

[10]

C. Kenig, G. Ponce and L. Vega, On the support of solutions to the generalized KdV equation,, Ann. Inst. H. Poincar\'e Anal. Non lin\'eaire, {19 (2002), 191. doi: 10.1016/S0294-1449(01)00073-7.

[11]

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

[12]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, (1983).

[13]

J. C. Saut and B. Scheurer, Unique continuation for some evolution equations,, J. Differential Equations, {66 (1987), 118. doi: 10.1016/0022-0396(87)90043-X.

[14]

H. Takaoka and N. Tzvetkov, On the local regularity of Kadomtsev-Petviashvili-II equation,, Internat. Math. Res. Notices, 2 (2001), 77. doi: 10.1155/S1073792801000058.

[1]

Michael Ruzhansky, Jens Wirth. Dispersive type estimates for fourier integrals and applications to hyperbolic systems. Conference Publications, 2011, 2011 (Special) : 1263-1270. doi: 10.3934/proc.2011.2011.1263

[2]

Yutian Lei. Wolff type potential estimates and application to nonlinear equations with negative exponents. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2067-2078. doi: 10.3934/dcds.2015.35.2067

[3]

Y. Efendiev, Alexander Pankov. Meyers type estimates for approximate solutions of nonlinear elliptic equations and their applications. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 481-492. doi: 10.3934/dcdsb.2006.6.481

[4]

Nakao Hayashi, Seishirou Kobayashi, Pavel I. Naumkin. Nonlinear dispersive wave equations in two space dimensions. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1377-1393. doi: 10.3934/cpaa.2015.14.1377

[5]

Jeremy L. Marzuola. Dispersive estimates using scattering theory for matrix Hamiltonian equations. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 995-1035. doi: 10.3934/dcds.2011.30.995

[6]

Yonggeun Cho, Tohru Ozawa, Suxia Xia. Remarks on some dispersive estimates. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1121-1128. doi: 10.3934/cpaa.2011.10.1121

[7]

Fabio Nicola. Remarks on dispersive estimates and curvature. Communications on Pure & Applied Analysis, 2007, 6 (1) : 203-212. doi: 10.3934/cpaa.2007.6.203

[8]

Peng Gao. Carleman estimates for forward and backward stochastic fourth order Schrödinger equations and their applications. Evolution Equations & Control Theory, 2018, 7 (3) : 465-499. doi: 10.3934/eect.2018023

[9]

Genni Fragnelli. Null controllability of degenerate parabolic equations in non divergence form via Carleman estimates. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 687-701. doi: 10.3934/dcdss.2013.6.687

[10]

Italo Capuzzo Dolcetta, Antonio Vitolo. Glaeser's type gradient estimates for non-negative solutions of fully nonlinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 539-557. doi: 10.3934/dcds.2010.28.539

[11]

Jerry Bona, Hongqiu Chen. Well-posedness for regularized nonlinear dispersive wave equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (4) : 1253-1275. doi: 10.3934/dcds.2009.23.1253

[12]

Hongqiu Chen, Jerry L. Bona. Periodic traveling--wave solutions of nonlinear dispersive evolution equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4841-4873. doi: 10.3934/dcds.2013.33.4841

[13]

Jerry L. Bona, Laihan Luo. More results on the decay of solutions to nonlinear, dispersive wave equations. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 151-193. doi: 10.3934/dcds.1995.1.151

[14]

El Mustapha Ait Ben Hassi, Farid Ammar khodja, Abdelkarim Hajjaj, Lahcen Maniar. Carleman Estimates and null controllability of coupled degenerate systems. Evolution Equations & Control Theory, 2013, 2 (3) : 441-459. doi: 10.3934/eect.2013.2.441

[15]

Victor Isakov. Carleman estimates for some anisotropic elasticity systems and applications. Evolution Equations & Control Theory, 2012, 1 (1) : 141-154. doi: 10.3934/eect.2012.1.141

[16]

Gong Chen, Peter J. Olver. Numerical simulation of nonlinear dispersive quantization. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 991-1008. doi: 10.3934/dcds.2014.34.991

[17]

Jerry Bona, Hongqiu Chen. Solitary waves in nonlinear dispersive systems. Discrete & Continuous Dynamical Systems - B, 2002, 2 (3) : 313-378. doi: 10.3934/dcdsb.2002.2.313

[18]

Ihyeok Seo. Carleman estimates for the Schrödinger operator and applications to unique continuation. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1013-1036. doi: 10.3934/cpaa.2012.11.1013

[19]

Matthias Eller, Daniel Toundykov. Carleman estimates for elliptic boundary value problems with applications to the stablization of hyperbolic systems. Evolution Equations & Control Theory, 2012, 1 (2) : 271-296. doi: 10.3934/eect.2012.1.271

[20]

D. Bartolucci, L. Orsina. Uniformly elliptic Liouville type equations: concentration compactness and a priori estimates. Communications on Pure & Applied Analysis, 2005, 4 (3) : 499-522. doi: 10.3934/cpaa.2005.4.499

2017 Impact Factor: 0.884

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]