American Institute of Mathematical Sciences

May  2012, 11(3): 1111-1127. doi: 10.3934/cpaa.2012.11.1111

Remarks on a two dimensional BBM type equation

 1 School of Mathematics and Computer Science, Damghan University, Damghan, Postal Code 36716--41167, Iran

Received  November 2010 Revised  August 2011 Published  December 2011

In this work, we study a two-dimensional version of the BBM equation. We prove that the Cauchy problem for this equation is globally well-posed in a natural space. We also show that the orbital stability of the solitary waves of the equation. Furthermore, we establish that if the solution of the Cauchy problem has a compact support for all times, then this solution vanishes identically.
Citation: Amin Esfahani. Remarks on a two dimensional BBM type equation. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1111-1127. doi: 10.3934/cpaa.2012.11.1111
References:
 [1] M. A. Abdou, The extended F-expansion method and its application for a class of nonlinear evolution equations,, Chaos, 31 (2007), 95. doi: 10.1016/j.chaos.2005.09.030. Google Scholar [2] M. J. Ablowitz and P. A. Clarkson, "Solitons, Nonlinear Evolution Equations and Inverse Scattering,'', Cambridge University Press (London Alath. Soc. Lecture Note Series 149), (1991). Google Scholar [3] H. Berestycki, T. Gallouët and O. Kavian, Équations de champs scalaires euclidiens non linéaires dans le plan,, C. R. Acad. Sci. Paris S\'er. I Math., 297 (1983), 307. Google Scholar [4] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations,, Arch. Rational Mech. Anal., 82 (1983), 313. doi: 10.1007/BF00250555. Google Scholar [5] J. L. Bona, Y. Liu and M. M. Tom, The Cauchy problem and stability of solitary-wave solutions for RLW-KP-type equations,, J. Differential Equations, 185 (2002), 437. doi: 10.1006/jdeq.2002.4171. Google Scholar [6] J. L. Bona, W. G. Pritchard and L. R. Scott, An evaluation of a model equation for water waves,, Philos. Trans. Royal Soc. London Series A, 302 (1981), 457. doi: 10.1098/rsta.1981.0178. Google Scholar [7] J. L. Bona and N. Tzvetkov, Sharp well-posedness results for the BBM equation,, Discrete Contin. Dyn. Syst., 23 (2009), 1241. doi: 10.3934/dcds.2009.23.1241. Google Scholar [8] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations,, Geom. Funct. Anal., 3 (1993), 107. doi: 10.1007/BF01896020. Google Scholar [9] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation,, Geom. Funct. Anal., 3 (1993), 209. doi: 10.1007/BF01895688. Google Scholar [10] J. Bourgain, On the Cauchy problem for the Kadomtsev-Petviashvili equation,, Geom. Funct. Anal., 3 (1993), 315. doi: 10.1007/BF01896259. Google Scholar [11] T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations,, Commun. Math. Phys., 85 (1982), 549. doi: 10.1007/BF01403504. Google Scholar [12] A. Constantin, Finite propagation speed for the Camassa-Holm equation,, J. Math. Phys., 46 (2005). doi: 10.1063/1.1845603. Google Scholar [13] A. de Bouard, Stability and instability of some nonlinear dispersive solitary waves in higher dimension,, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 89. doi: 10.1017/S0308210500030614. Google Scholar [14] A. V. Faminskii, The Cauchy problem for the Zakharov-Kuznetsov equation,, J. Differ. Equ., 31 (1995), 1002. Google Scholar [15] J. Ginibre, Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace (d'après Bourgain),, Ast\'erisque, 4 (1996), 163. Google Scholar [16] M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. I,, J. Funct. Anal., 74 (1987), 160. doi: 10.1016/0022-1236(87)90044-9. Google Scholar [17] J. Hammack and H. Segur, The Kortweg-de Vries equation and water waves. II. Comparison with experiments,, J. Fluid Mech., 65 (1974), 289. doi: 10.1017/S002211207400139X. Google Scholar [18] M. A. Johnson, The transverse instability of periodic waves in Zakharov-Kuznetsov type equations,, Studies in Applied Mathematics, 124 (2010), 323. doi: 10.1111/j.1467-9590.2009.00473.x. Google Scholar [19] F. Linares and A. Pastor, Well-posedness for the two-dimensional modified Zakharov-Kuznetsov equation,, SIAM J. Math. Anal., 41 (2009), 1323. doi: 10.1137/080739173. Google Scholar [20] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I.,, Ann. Inst. H. Poincar\'e, 1 (1984), 109. Google Scholar [21] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II.,, Ann. Inst. H. Poincar\'e, 4 (1984), 223. Google Scholar [22] Y. Mammeri, Unique continuation property for the KP-BBM-II equation,, Differential Integral Equations, 22 (2009), 393. Google Scholar [23] L. Molinet, On the asymptotic behavior of solutions to the (generalized) Kadomtsev-Petviashvili-Burgers equations,, J. Differential Equations, 152 (1999), 30. doi: 10.1006/jdeq.1998.3522. Google Scholar [24] M. Panthee, A note on the unique continuation property for Zakharov-Kuznetsov equation,, Nonlinear Anal., 29 (2004), 425. doi: 10.1016/j.na.2004.07.022. Google Scholar [25] S. I. Pohozaev, On the eigenfunctions of the equation $\Delta u+\lambda f(u)$,, Dokl. Akad. Nauk SSSR, 165 (1965), 36. Google Scholar [26] J.-C. Saut and N. Tzvetkov, Global well-posedness for the KP-BBM equations,, Appl. Math. Res. Express, 1 (2004), 1. doi: 10.1155/S1687120004010718. Google Scholar [27] J.-C. Saut and N. Tzvetkov, On periodic KP-I type equations,, Comm. Math. Phys., 221 (2001), 451. doi: 10.1007/PL00005577. Google Scholar [28] S. K. Turitsyn, J. J. Rasmussen and M. A. Raadu, "Stability of Weak Double Layers,'', Royal Institute of Technology, (1991), 91. Google Scholar [29] A. M. Wazwaz, Compact and noncompact physical structures for the ZK-BBM equation,, Appl. Math. Comput., 169 (2005), 713. doi: 10.1016/j.amc.2004.09.062. Google Scholar [30] N. J. Zabusky and C. Galvin, Shallow-water waves, the Korteweg-de Vries equation and solitons,, J. Fluid Mech., 47 (1971), 811. Google Scholar [31] V. E. Zakharov and E. A. Kuznetsov, Three dimensional solitons,, Sov. Phys. JETP, 39 (1974), 285. Google Scholar

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References:
 [1] M. A. Abdou, The extended F-expansion method and its application for a class of nonlinear evolution equations,, Chaos, 31 (2007), 95. doi: 10.1016/j.chaos.2005.09.030. Google Scholar [2] M. J. Ablowitz and P. A. Clarkson, "Solitons, Nonlinear Evolution Equations and Inverse Scattering,'', Cambridge University Press (London Alath. Soc. Lecture Note Series 149), (1991). Google Scholar [3] H. Berestycki, T. Gallouët and O. Kavian, Équations de champs scalaires euclidiens non linéaires dans le plan,, C. R. Acad. Sci. Paris S\'er. I Math., 297 (1983), 307. Google Scholar [4] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations,, Arch. Rational Mech. Anal., 82 (1983), 313. doi: 10.1007/BF00250555. Google Scholar [5] J. L. Bona, Y. Liu and M. M. Tom, The Cauchy problem and stability of solitary-wave solutions for RLW-KP-type equations,, J. Differential Equations, 185 (2002), 437. doi: 10.1006/jdeq.2002.4171. Google Scholar [6] J. L. Bona, W. G. Pritchard and L. R. Scott, An evaluation of a model equation for water waves,, Philos. Trans. Royal Soc. London Series A, 302 (1981), 457. doi: 10.1098/rsta.1981.0178. Google Scholar [7] J. L. Bona and N. Tzvetkov, Sharp well-posedness results for the BBM equation,, Discrete Contin. Dyn. Syst., 23 (2009), 1241. doi: 10.3934/dcds.2009.23.1241. Google Scholar [8] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations,, Geom. Funct. Anal., 3 (1993), 107. doi: 10.1007/BF01896020. Google Scholar [9] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation,, Geom. Funct. Anal., 3 (1993), 209. doi: 10.1007/BF01895688. Google Scholar [10] J. Bourgain, On the Cauchy problem for the Kadomtsev-Petviashvili equation,, Geom. Funct. Anal., 3 (1993), 315. doi: 10.1007/BF01896259. Google Scholar [11] T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations,, Commun. Math. Phys., 85 (1982), 549. doi: 10.1007/BF01403504. Google Scholar [12] A. Constantin, Finite propagation speed for the Camassa-Holm equation,, J. Math. Phys., 46 (2005). doi: 10.1063/1.1845603. Google Scholar [13] A. de Bouard, Stability and instability of some nonlinear dispersive solitary waves in higher dimension,, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 89. doi: 10.1017/S0308210500030614. Google Scholar [14] A. V. Faminskii, The Cauchy problem for the Zakharov-Kuznetsov equation,, J. Differ. Equ., 31 (1995), 1002. Google Scholar [15] J. Ginibre, Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace (d'après Bourgain),, Ast\'erisque, 4 (1996), 163. Google Scholar [16] M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. I,, J. Funct. Anal., 74 (1987), 160. doi: 10.1016/0022-1236(87)90044-9. Google Scholar [17] J. Hammack and H. Segur, The Kortweg-de Vries equation and water waves. II. Comparison with experiments,, J. Fluid Mech., 65 (1974), 289. doi: 10.1017/S002211207400139X. Google Scholar [18] M. A. Johnson, The transverse instability of periodic waves in Zakharov-Kuznetsov type equations,, Studies in Applied Mathematics, 124 (2010), 323. doi: 10.1111/j.1467-9590.2009.00473.x. Google Scholar [19] F. Linares and A. Pastor, Well-posedness for the two-dimensional modified Zakharov-Kuznetsov equation,, SIAM J. Math. Anal., 41 (2009), 1323. doi: 10.1137/080739173. Google Scholar [20] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I.,, Ann. Inst. H. Poincar\'e, 1 (1984), 109. Google Scholar [21] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II.,, Ann. Inst. H. Poincar\'e, 4 (1984), 223. Google Scholar [22] Y. Mammeri, Unique continuation property for the KP-BBM-II equation,, Differential Integral Equations, 22 (2009), 393. Google Scholar [23] L. Molinet, On the asymptotic behavior of solutions to the (generalized) Kadomtsev-Petviashvili-Burgers equations,, J. Differential Equations, 152 (1999), 30. doi: 10.1006/jdeq.1998.3522. Google Scholar [24] M. Panthee, A note on the unique continuation property for Zakharov-Kuznetsov equation,, Nonlinear Anal., 29 (2004), 425. doi: 10.1016/j.na.2004.07.022. Google Scholar [25] S. I. Pohozaev, On the eigenfunctions of the equation $\Delta u+\lambda f(u)$,, Dokl. Akad. Nauk SSSR, 165 (1965), 36. Google Scholar [26] J.-C. Saut and N. Tzvetkov, Global well-posedness for the KP-BBM equations,, Appl. Math. Res. Express, 1 (2004), 1. doi: 10.1155/S1687120004010718. Google Scholar [27] J.-C. Saut and N. Tzvetkov, On periodic KP-I type equations,, Comm. Math. Phys., 221 (2001), 451. doi: 10.1007/PL00005577. Google Scholar [28] S. K. Turitsyn, J. J. Rasmussen and M. A. Raadu, "Stability of Weak Double Layers,'', Royal Institute of Technology, (1991), 91. Google Scholar [29] A. M. Wazwaz, Compact and noncompact physical structures for the ZK-BBM equation,, Appl. Math. Comput., 169 (2005), 713. doi: 10.1016/j.amc.2004.09.062. Google Scholar [30] N. J. Zabusky and C. Galvin, Shallow-water waves, the Korteweg-de Vries equation and solitons,, J. Fluid Mech., 47 (1971), 811. Google Scholar [31] V. E. Zakharov and E. A. Kuznetsov, Three dimensional solitons,, Sov. Phys. JETP, 39 (1974), 285. Google Scholar
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