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Sharp global well-posedness of the BBM equation in $L^p$ type Sobolev spaces

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  • The global well-posedness of the BBM equation is established in $H^{s,p}(\textbf{R})$ with $s\geq \max\{0,\frac{1}{p}-\frac{1}{2}\}$ and $1\leq p<\infty$. Moreover, the well-posedness results are shown to be sharp in the sense that the solution map is no longer $C^2$ from $H^{s,p}(\textbf{R})$ to $C([0,T];H^{s,p}(\textbf{R}))$ for smaller $s$ or $p$. Finally, some growth bounds of global solutions in terms of time $T$ are proved.
    Mathematics Subject Classification: Primary: 35Q53; Secondary: 35A01.

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  • [1]

    J. Avrin and J. A. Goldstein, Global existence for the Benjamin-Bona-Mahony equation in arbitrary dimensions, Nonlinear Analysis, 9 (1985), 861-865.doi: 10.1016/0362-546X(85)90023-9.

    [2]

    J. Avrin, The generalized Benjamin-Bona-Mahony equation in $R^n$ with singular initial data, Nonlinear Analysis, 11 (1987), 139-147.doi: 10.1016/0362-546X(87)90032-0.

    [3]

    H. Bae and A. Biswas, Gevrey regularity for a class of dissipative equations with analytic nonlinearity, Methods Appl. Anal., 22 (2015), 377-408.doi: 10.4310/MAA.2015.v22.n4.a3.

    [4]

    T. Benjamin, J. Bona and J. Mahony, Model equations for long waves in nonlinear dispersive systems, Phil.Trans. R. Soc., 272 (1972), 47-78.doi: 10.1098/rsta.1972.0032.

    [5]

    J. L. Bona and N. Tzvetkov, Sharp well-posedness results for the BBM equation, Discrete Contin. Dyn. Syst, 23 (2009), 1241-1252.doi: 10.3934/dcds.2009.23.1241.

    [6]

    J. Bourgain, On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE, International Mathematics Research Notices, 6 (1996), 277-304.doi: 10.1155/S1073792896000207.

    [7]

    X. Carvajal and M. Panthee, On ill-posedness for the generalized BBM equation, Discrete and Continuous Dynamical Systems - Series A, 34 (2014), 4565-4576.doi: 10.3934/dcds.2014.34.4565.

    [8]

    W. Chen, Z. Guo and J. Xiao, Sharp well-posedness for the Benjamin equation, Nonlinear Analysis, 74 (2011), 6209-6230.doi: 10.1016/j.na.2011.06.002.

    [9]

    J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $R$ and $T$, Journal of the American Mathematical Society, 16 (2003), 705-749.doi: 10.1090/S0894-0347-03-00421-1.

    [10]

    Q. Deng, Y. Ding and X. Yao, Gaussian bounds for higher-order elliptic differential operators with Kato type potentials, J. Funct. Anal., 266 (2014), 5377-5397.doi: 10.1016/j.jfa.2014.02.014.

    [11]

    J. A. Goldstein and B. J. Wichnoski, On the Benjamin-Bona-Mahony equation in higher dimensions, Nonlinear Analysis, 4 (1980), 665-675.doi: 10.1016/0362-546X(80)90067-X.

    [12]

    L. Grafakos, Modern Fourier Analysis, Spinger 2nd ed., New York, 2009.doi: 10.1007/978-1-4939-1230-8.

    [13]

    L. Grafakos and S. Oh, The Kato-Ponce Inequality, Communications in Partial Differential Equations, 39 (2014), 1128-1157.doi: 10.1080/03605302.2013.822885.

    [14]

    Y. Guo, M. Wang and Y. Tang, Higher regularity of global attractor for a damped Benjamin-Bona-Mahony equation on R, Applicable Analysis: An International Journal, 94 (2015), 1766-1783.doi: 10.1080/00036811.2014.946561.

    [15]

    C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized korteweg-de vries equation via the contraction principle, Communications on Pure and Applied Mathematics, 46 (1993), 527-620.doi: 10.1002/cpa.3160460405.

    [16]

    Y. Li and Y. Wu, Global well-posedness for the Benjamin eqaution in low regularity, Nonlinear Anal., 73 (2010), 1610-1625.doi: 10.1016/j.na.2010.04.068.

    [17]

    J. Nahas, A Decay Property of Solutions to the mKdV Equation, PhD. Thesis University of California-Santa Barbara, June 2010.

    [18]

    P. J. Olver, Euler operators and conservation laws of the BBM equation, Mathematical Proceedings of the Cambridge Philosophical Society, 85 (1979), 143-160.doi: 10.1017/S0305004100055572.

    [19]

    M. Panthee, On the ill-posedness result for the BBM equation, Discrete Contin. Dyn. Syst., 30 (2011), 253-259.doi: 10.3934/dcds.2011.30.253.

    [20]

    M. Reed, B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-adjointness, Elsevier, 1975.

    [21]

    D. Roumégoux, A symplectic non-squeezing theorem for BBM equation, Dyn. Partial Differ. Equ., 7 (2010), 289-305.doi: 10.4310/DPDE.2010.v7.n4.a1.

    [22]

    W. Rudin, Functional Analysis, International series in pure and applied mathematics, 1991.

    [23]

    V. Sohinger, Bounds on the Growth of High Sobolev Norms of Solutions to Nonlinear Schrödinger Equations, PhD. Thesis, Massachusetts Institute of Technology, April 2011.

    [24]

    E. M. Stein, Singular Integral and Differential Property of Functions, New Jersey: Princeton Univ. Press, 1970.

    [25]

    H. Wang and S. Cui, Global well-posedness of the Cauchy problem of the fifth-order shallow water equation, Journal of Differential Equations, 230 (2006), 600-613.doi: 10.1016/j.jde.2006.04.008.

    [26]

    M. Wang, Long time dynamics for a damped Benjamin-Bona-Mahony equation in low regularity spaces, Nonlinear Analysis, 105 (2014), 134-144.doi: 10.1016/j.na.2014.04.013.

    [27]

    M. Wang, Long time behavior of a damped generalized BBM equation in low regularity spaces, Mathematical Methods in the Applied Sciences, 38 (2015), 4852-4866.doi: 10.1002/mma.3400.

    [28]

    X. Yang and Y. Li, Global well-posedness for a fifth-order shallow water equation in Sobolev spaces, Journal of Differential Equations, 248 (2010), 1458-1472.doi: 10.1016/j.jde.2010.01.004.

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