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

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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

Pedro Isaza ^1, and Jorge Mejía ^1,

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

Received February 2010 Revised October 2010 Published April 2011

Abstract
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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.

Keywords: Nonlinear dispersive equations, estimates of Carleman type..

Mathematics Subject Classification: Primary: 35Q53; Secondary: 37K0.

Citation: Pedro Isaza, Jorge Mejía. On the support of solutions to the Kadomtsev-Petviashvili (KP-II) equation. Communications on Pure and 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), 437-447. 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), 504-535. 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), 957-971.
[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), 673-708. 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. Poincaré Anal. Non Linéaire, 26 (2009), 917-941. 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), 1027-1054. 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), 1-12.
[8]	P. Isaza and J. Mejía, On the support of solutions to the Ostrovsky equation with negative dispersion, J. Differential Equations, 247 (2009), 1851-1865. 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), 4016-4029. 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. Poincaré Anal. Non linéaire, 19 (2002), 191-208. 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.
[12]	A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.
[13]	J. C. Saut and B. Scheurer, Unique continuation for some evolution equations, J. Differential Equations, 66 (1987), 118-139. 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), 77-114. 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), 437-447. 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), 504-535. 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), 957-971.
[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), 673-708. 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. Poincaré Anal. Non Linéaire, 26 (2009), 917-941. 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), 1027-1054. 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), 1-12.
[8]	P. Isaza and J. Mejía, On the support of solutions to the Ostrovsky equation with negative dispersion, J. Differential Equations, 247 (2009), 1851-1865. 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), 4016-4029. 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. Poincaré Anal. Non linéaire, 19 (2002), 191-208. 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.
[12]	A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.
[13]	J. C. Saut and B. Scheurer, Unique continuation for some evolution equations, J. Differential Equations, 66 (1987), 118-139. 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), 77-114. doi: 10.1155/S1073792801000058.

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