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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[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.  Google Scholar

[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.  Google Scholar

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