Well-posedness and unique continuation property for the solutions to the generalized Kawahara equation below the energy space

References

• Kawahara T . Oscillatory solitary waves in dispersive media. J Phys Soc Japan. 1972;33:260–264.
   Google Scholar
• Colliander J , Keel M , Stalfilanti G , et al . Global wel-posedness for the KdV in Sobolev spaces of negative indices. EJDE. 2001;26:1–7.
   Google Scholar
• Colliander J , Keel M , Stalfilanti G , et al . Sharp global wel-posedness for the KdV and modified KdV on R and T. J Amer Math Soc. 2003;16:705–749.
   Web of Science ®Google Scholar
• Kenig CE , Ponce G , Vega L . Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle. Commun Pure Appl Math. 1993;46:527–620.
   Web of Science ®Google Scholar
• Kenig CE , Ponce G , Vega L . The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices. Duke Math J. 1993;71:1–21.
   Web of Science ®Google Scholar
• Kenig CE , Ponce G , Vega L . A bilinear estimate with application to the KdV equation. J Amer Math Soc. 1996;9:573–603.
   Web of Science ®Google Scholar
• Molinet L , Ribaud F . On the Cauchy problem for the generalized Korteweg-de Vries equation. Commun Partial Differ Equ. 2003;28:2065–2091.
   Web of Science ®Google Scholar
• Guo ZH . Global well-posedness of the Korteweg-de Vries equation in H -3/4(R). J Math Pure Appl. 2009;9:583–597.
   Web of Science ®Google Scholar
• Cui SB , Deng DG , Tao SP . Global existence of solutions for the Cauchy problem of the Kawahara equation with L 2 initial data. Acta Math Sinica Eng Ser. 2006;22:1457–1466.
   Web of Science ®Google Scholar
• Cui SB , Tao SP . Strichartz estimates for dispersive equations and sovability of Cauchy problems of the Kawahara equation. J Math Anal Appl. 2005;304:683–702.
   Web of Science ®Google Scholar
• Wang H , Cui SB , Deng DG . Global existence of solutions of the Kawahara equation in Sobolev spaces of negative indices. Acta Math Sinica Eng Ser. 2007;23:1435–1446.
   Web of Science ®Google Scholar
• Chen WG , Li JF , Miao CX , et al. Low regularity solutions of two fifth-order KdV type equations. J Anal Math. 2009:221–238.
   Web of Science ®Google Scholar
• Jia YL , Huo ZH . Well-posedness of the fifth-order shallow water equations. J Differ Equ. 2009;246:2448–2467.
   Web of Science ®Google Scholar
• Huo ZH . The Cauchy problem for the fifth-order shallow water equations. Acta Math Sinica Eng Ser. 2005;21:441–454.
   Web of Science ®Google Scholar
• Tao SP , Cui SB . Laocal and global existence of solutions to initial value probles of modified nonlinear Kawahara equations. Acta Math Sinica Eng Ser. 2005;21:1035–1044.
   Web of Science ®Google Scholar
• Yan W , Li YS . The Cauchy problem for Kawahara equation in Sobolev spaces with low regularity. Math Meth Appl Sci. 2010;33:1647–1660.
   Web of Science ®Google Scholar
• Yan W , Li YS , Yang XY . The Cauchy problem for the modified Kawahara equation in Sobolev spaces with low regularity. Math Comput Model. 2011;54:1251–1261.
   Google Scholar
• Chen WG , Guo ZH . Global well-posedness and I-method for the fifth order Korteweg-de Vries equation. J Anal Math. 2011;114:121–156.
   Web of Science ®Google Scholar
• Hunter JK , Scheurle J . Existence of perturbed solitary wave solutions to a model equation for wave solutions to a model equation for water waves. Phys D. 1988;32:253–268.
   Web of Science ®Google Scholar
• Kichenassamy S , Olver PJ . Existence and nonexistence of solitary wave solutions to higher-order model evolution equations. SIAM J Math Anal. 1992;23:1141–1166.
   Web of Science ®Google Scholar
• Sun SM . Existence of a generalized solitary wave with positive Bond number smaller than 1/3. J Math Anal Appl. 1991;156:471–504.
   Web of Science ®Google Scholar
• Olver PJ . Applications of Lie groups to differential equations. 2nd ed. New York (NY): Springer-Verlag; 1993.
   Google Scholar
• Carleman T . Sur les systémes Lineares aux dérivées partielles du premier ordre á deux variables. C R Acad Sci Paris. 1939;97:471–474.
   Google Scholar
• Saut JC , Scheurer B . Unique contiuation for some evolution equations. J Differ Equ. 1987;66:118–139.
   Web of Science ®Google Scholar
• Isakov V . Carleman type estimates in an anisotropic case and applications. J Differ Equ. 1993;105:217–238.
   Web of Science ®Google Scholar
• Zhang BY . Unique contiuation for the Korteweg-de Vries equation. SIAM J Math Anal. 1992;23:55–71.
   Web of Science ®Google Scholar
• Bourgain J . On the compactness of the support of solutions of dispersive equations. Internat Math Res Not. 1997;9:437–447.
   Google Scholar
• Carvajal X , Panthee M . Unique contiuation property for a higher order nonlinear Schrödinger equation. J Math Anal Appl. 2005;33:188–207.
   Web of Science ®Google Scholar
• Kenig CE , Ponce G , Vega L . On the support of solutions to the generalized KdV equation. Ann Inst H Poincaré Anal Non Linéaire. 2002;19:191–208.
   Web of Science ®Google Scholar
• Silva PN . Unique contiuation for the Kawahara equation. Tend Math Appl Comput. 2007;8:463–473.
   Google Scholar
• Escauriaza L , Kenig CE , Ponce G , et al . On uniqueness properties of solutions of the k-generalized KdV equations. J Funct Anal. 2007;244:504–535.
   Web of Science ®Google Scholar
• Escauriaza L , Kenig CE , Ponce G , et al . Convexity properties of solutions to the free Schrödinger equation with Gaussian decay. Math Res Lett. 2008;15:957–972.
   Web of Science ®Google Scholar
• Escauriaza L , Kenig CE , Ponce G , et al . On unique continuation of solutions of Convexity properties of solutions to the free Schrödinge equations. Commun Partial Differ Equ. 2006;31:1811–1823.
   Web of Science ®Google Scholar
• Isaza P , Mejía J . On the support of solutions to the Ostrovsky equation with negative dispersion. J Differ Equ. 2009;247:1851–1865.
   Web of Science ®Google Scholar
• Isaza P , Mejía J . On the support of solutions to the Ostrovsky equation with positive dispersion. Nonlinear Anal. 2010;72:4016–4029.
   Web of Science ®Google Scholar
• Bustamante E , Isaza P , Mejía J . On uniqueness properties of solutions of the Ostrovsky equation with positive dispersion. Nonlinear Anal. 2012;75:5523–5540.
   Web of Science ®Google Scholar
• Isaza P . Unique continuation principle for the Ostrovsky equation with negative dispersion. J Differ Equ. 2013;255:796–811.
   Web of Science ®Google Scholar
• Tao T . Multilinear weighted convolution of L 2 functions, and applications to nonlinear dispersive equations. Amer J Math. 2001;123:839–908.
   Web of Science ®Google Scholar
• Sjoberg A . On the Korteweg-de Vries equation: existence and uniqueness. J Math Anal Appl. 1970;29:569–579.
   Web of Science ®Google Scholar
• Holden H , Koley U , Risebro NH . Convergence of a fully discrete finite difference scheme for the Korteweg-de Vries equation. IMA J Numer Anal. 2015;35(3):1047–1077.
   Web of Science ®Google Scholar
• Dutta R , Koley U , Risebro NH . Convergence of a higher order scheme for the Korteweg-de Vries equation. SIAM J Numer Anal. 2015;53(4):1963–1983.
   Web of Science ®Google Scholar
• Koley U . Finite difference schemes for the Korteweg-de Vries-Kawahara equation. Int J Numer Anal Model. 2016;13(3):344–367.
   Web of Science ®Google Scholar
• Tian LX , Gui GL , Liu Y . On the Cauchy problem for the generalized shallow water wave equation. J Differ Equ. 2008;245:1838–1852.
   Web of Science ®Google Scholar
• Kenig CE , Ponce G , Vega L . Oscillatory integrals and regularity of dispersive equations. Iniana Univ Math J. 1991;40:33–69.
   Web of Science ®Google Scholar
• Ginibre J , Tsutsumi Y , Velo Y . On the Cauchy problem for the Zakharov system. J Funct Anal. 1997;151:384–436.
   Web of Science ®Google Scholar
• Zhang ZY , Huang JH . Well-posedness and unique continuation property for the generalized Ostrovsky equation with low regularity. Math Meth Appl Sci. 2016;39:2488–2513.
   Web of Science ®Google Scholar
• Zhang ZY , Huang JH , Liu ZH , et al . On the unique continuation property for the modified Kawahara equation. Adv Math (China). 2016;45(1):80–88.
   Google Scholar
• Zhang ZY , Huang JH , Sun MB . Almost conservation laws and global rough solutions of the defocusing nonlinear wave equation on R 2 . Acta Math Sci. 2017;37(2):385–394.
   Web of Science ®Google Scholar
• Linares F , Milanés A . Local and global well-posedness for the Ostrovsky equation. J Differ Equ. 2006;222:325–340.
   Web of Science ®Google Scholar