# American Institute of Mathematical Sciences

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

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

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 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.
 [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 and Continuous Dynamical Systems, 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 and 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 and 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 and Continuous Dynamical Systems, 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 and 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 and 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 and 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 and Continuous Dynamical Systems - S, 2013, 6 (3) : 687-701. doi: 10.3934/dcdss.2013.6.687 [10] Jerry Bona, Hongqiu Chen. Well-posedness for regularized nonlinear dispersive wave equations. Discrete and Continuous Dynamical Systems, 2009, 23 (4) : 1253-1275. doi: 10.3934/dcds.2009.23.1253 [11] Hongqiu Chen, Jerry L. Bona. Periodic traveling--wave solutions of nonlinear dispersive evolution equations. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 4841-4873. doi: 10.3934/dcds.2013.33.4841 [12] Fabrício Cristófani, Ademir Pastor. Nonlinear stability of periodic-wave solutions for systems of dispersive equations. Communications on Pure and Applied Analysis, 2020, 19 (10) : 5015-5032. doi: 10.3934/cpaa.2020225 [13] Jerry L. Bona, Laihan Luo. More results on the decay of solutions to nonlinear, dispersive wave equations. Discrete and Continuous Dynamical Systems, 1995, 1 (2) : 151-193. doi: 10.3934/dcds.1995.1.151 [14] Italo Capuzzo Dolcetta, Antonio Vitolo. Glaeser's type gradient estimates for non-negative solutions of fully nonlinear elliptic equations. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 539-557. doi: 10.3934/dcds.2010.28.539 [15] Oana Ivanovici. Dispersive estimates for the wave and the Klein-Gordon equations in large time inside the Friedlander domain. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5707-5742. doi: 10.3934/dcds.2021093 [16] El Mustapha Ait Ben Hassi, Farid Ammar khodja, Abdelkarim Hajjaj, Lahcen Maniar. Carleman Estimates and null controllability of coupled degenerate systems. Evolution Equations and Control Theory, 2013, 2 (3) : 441-459. doi: 10.3934/eect.2013.2.441 [17] Victor Isakov. Carleman estimates for some anisotropic elasticity systems and applications. Evolution Equations and Control Theory, 2012, 1 (1) : 141-154. doi: 10.3934/eect.2012.1.141 [18] Gong Chen, Peter J. Olver. Numerical simulation of nonlinear dispersive quantization. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 991-1008. doi: 10.3934/dcds.2014.34.991 [19] Jerry Bona, Hongqiu Chen. Solitary waves in nonlinear dispersive systems. Discrete and Continuous Dynamical Systems - B, 2002, 2 (3) : 313-378. doi: 10.3934/dcdsb.2002.2.313 [20] Chenjie Fan, Zehua Zhao. Decay estimates for nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2021, 41 (8) : 3973-3984. doi: 10.3934/dcds.2021024

2020 Impact Factor: 1.916

## Metrics

• HTML views (0)
• Cited by (4)

• on AIMS