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A non-homogeneous boundary value problem of the sixth order Boussinesq equation in a quarter plane. (English) Zbl 1397.35257

Summary: The paper is concerned with an initial-boundary-value problem of the sixth order Boussinesq equation posed on a quarter plane with non-homogeneous boundary conditions: \[ \begin{cases} u_{tt}-u_{xx}+\beta u_{xxxx}-u_{xxxxxx}+(u^2)_{xx}=0,\quad\text{for}\, x>0, t>0,\\ u(x,0)=\varphi(x),ut(x,0)=\psi^{\prime\prime}(x),\\ u(0,t)=h_1(t),u_{xx}(0,t)=h_2(t),u_{xxxx}(0,t)=h_3(t),\end{cases} \eqno{(1)} \] where \(\beta=\pm1\). It is shown that the problem is locally well-posed in the space \(H^s(\mathbb{R}^+)\) for any \(0\leq s<\frac{13}{2}\) with the initial data \((\varphi,\psi)\) in the space \[ H^s(\mathbb{R}^+)\times H^{s-1}(\mathbb{R}^+) \] and the naturally compatible boundary data \[ h_1\in H^{\frac{s+1}{3}}_{\operatorname{loc}}(\mathbb{R}^+), h_2\in H^{\frac{s-1}{3}}_{\operatorname{loc}}(\mathbb{R}^+)\;\text{and}\; h_3\in H^{\frac{s-3}{3}}_{\operatorname{loc}}(\mathbb{R}^+) \] with optimal regularity.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35Q55 NLS equations (nonlinear Schrödinger equations)
35Q35 PDEs in connection with fluid mechanics
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[1] J. Bergh and J. Lofstrom, Interpolation Spaces: An Introduction, Springer-Verlag Berlin Heidelberg, New York, 1976. · Zbl 0344.46071
[2] J. L. Bona; M. Chen, A boussinesq system for two-way propagation of nonlinear dispersive waves, Physica D: Nonlinear Phenomena, 116, 191-224 (1998) · Zbl 0962.76515 · doi:10.1016/S0167-2789(97)00249-2
[3] J. L. Bona; M. Chen; J. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. ii. the nonlinear theory, Nonlinearity, 17, 925-952 (2004) · Zbl 1059.35103 · doi:10.1088/0951-7715/17/3/010
[4] J. L. Bona; M. Chen; J. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. i. derivation and linear theory, J. Nonlinear Sci., 12, 283-318 (2002) · Zbl 1022.35044 · doi:10.1007/s00332-002-0466-4
[5] J. L. Bona; R. L. Sachs, Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation, Comm. Math. Phys., 118, 15-29 (1998) · Zbl 0654.35018 · doi:10.1007/BF01218475
[6] J. L. Bona; S. M. Sun; B.-Y. Zhang, A non-homogeneous boundary-value problem for the Korteweg-de Vries equation in a quarter plane, Trans. Amer, Math. Soc., 354, 427-490 (2002) · Zbl 0988.35141 · doi:10.1090/S0002-9947-01-02885-9
[7] J. L. Bona; S. M. Sun; B.-Y. Zhang, A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain, Comm. Partial Differential Equations, 28, 1391-1436 (2003) · Zbl 1057.35049 · doi:10.1081/PDE-120024373
[8] J. L. Bona; S. M. Sun; B.-Y. Zhang, A non-homogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain. II, J. Differential Equations, 247, 2558-2596 (2009) · Zbl 1181.35228 · doi:10.1016/j.jde.2009.07.010
[9] J. L. Bona, S. M. Sun and B. -Y. Zhang, Nonhomo Boundary-value problems for Onedimensional nonlinear Schrodinger equations, J. Math. Pures Appl., 109 (2018), 1-66, arXiv: 1503.00065, [math.AP].
[10] J. Boussinesq, Theorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal equation, J. Math. Pures Appl., 17, 55-108 (1872) · JFM 04.0493.04
[11] C. Christov; G. Maugin; M. Velarde, Well-posed Boussinesq paradigm with purely spatial higher-order derivatives, Phys. Rev. E, 54, 3621-3638 (1996) · doi:10.1103/PhysRevE.54.3621
[12] J. E. Colliander; C. E. Kenig, The generalized Korteweg-de Vries equation on the half line, Comm. Partial Differential Equations, 27, 2187-2266 (2002) · Zbl 1041.35064 · doi:10.1081/PDE-120016157
[13] J. de Frutos; T. Ortega; J. M. Sanz-Serna, Pseudospectral method for the “good” Boussinesq equation, Math. Comp., 57, 109-122 (1991) · Zbl 0735.65089
[14] A. Esfahani; L. G. Farah, Local well-posedness for the sixth-order Boussinesq equation, Journal of Mathematical Analysis and Applications, 385, 230-242 (2012) · Zbl 1231.35045 · doi:10.1016/j.jmaa.2011.06.038
[15] A. Esfahani; L. G. Farah; H. Wang, Global existence and blow-up for the generalized sixth-order Boussinesq equation, Nonlinear Anal., 75, 4325-4338 (2012) · Zbl 1246.35046 · doi:10.1016/j.na.2012.03.019
[16] A. Esfahani; H. Wang, A bilinear estimate with application to the sixth-order Boussinesq equation, Differential Integral Equations, 27, 401-414 (2014) · Zbl 1340.35258
[17] Y.-F. Fang; M. G. Grillakis, Existence and uniqueness for Boussinesq type equations on a circle, Comm.Partial Differential Equations, 21, 1253-1277 (1996) · Zbl 0859.35096 · doi:10.1080/03605309608821225
[18] L. G. Farah, Local solutions in Sobolev spaces with negative indices for the “good” Boussinesq equation, Comm. Partial Differential Equations, 34, 52-73 (2009) · Zbl 1173.35682 · doi:10.1080/03605300802682283
[19] L. G. Farah; M. Scialom, On the periodic “good ” Boussinesq equation, Proc. Amer. Math. Soc., 138, 953-964 (2010) · Zbl 1196.35145 · doi:10.1090/S0002-9939-09-10142-9
[20] B.-F. Feng; T. Kawahara; T. Mitsui; Y.-S. Chan, Solitary-wave propagation and interactions for a sixth-order generalized Boussinesq equation, Int. J. Math. Math. Sci., 2005, 1435-1448 (2005) · Zbl 1081.35086
[21] J. Holmer, The initial-boundary-value problem for the 1d nonlinear schr{ö}dinger equation on the half-line, Differential and Integral equations, 18, 647-668 (2005) · Zbl 1212.35448
[22] R. Hunt; Muckenhoupt; W. Benjamin; R. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc., 176, 227-251 (1973) · Zbl 0262.44004 · doi:10.1090/S0002-9947-1973-0312139-8
[23] O. Kamenov, Exact periodic solutions of the sixth-order generalized Boussinesq equation, J. Phys. A, 42 (2009), 375501, 11 pp. · Zbl 1180.35042
[24] C. E. Kenig; G. Ponce; L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46, 527-620 (1993) · Zbl 0808.35128 · doi:10.1002/cpa.3160460405
[25] C. E. Kenig; G. Ponce; L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. l Soc., 9, 573-603 (1996) · Zbl 0848.35114 · doi:10.1090/S0894-0347-96-00200-7
[26] F. Linares, Global existence of small solutions for a generalized Boussinesq equation, J. Differential Equations, 106, 257-293 (1993) · Zbl 0801.35111 · doi:10.1006/jdeq.1993.1108
[27] J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, volume 1. Die Grundlehren der mathematischen Wissenschaften, Band 182. Springer-Verlag, New York-Heidelberg, 1972. · Zbl 0227.35001
[28] F.-L. Liu; D. L. Russell, Solutions of the Boussinesq equation on a periodic domain, J. Math. Anal. Appl., 194, 78-102 (1995) · Zbl 0833.35113 · doi:10.1006/jmaa.1995.1287
[29] Y. Liu, Instability of solitary waves for generalized Boussinesq equations, J. Dynam. Differential Equations, 5, 537-558 (1993) · Zbl 0784.34048 · doi:10.1007/BF01053535
[30] Y. Liu, Instability and blow-up of solutions to a generalized Boussinesq equation, SIAM J. Math. Anal., 26, 1527-1546 (1995) · Zbl 0857.35103 · doi:10.1137/S0036141093258094
[31] Y. Liu, Decay and scattering of small solutions of a generalized Boussinesq equation, J. Funct. Anal., 147, 51-68 (1997) · Zbl 0884.35129 · doi:10.1006/jfan.1996.3052
[32] Y. Liu, Strong instability of solitary-wave solutions of a generalized Boussinesq equation, J. Differential Equations, 164, 223-239 (2000) · Zbl 0973.35163 · doi:10.1006/jdeq.2000.3765
[33] G. A. Maugin, Nonlinear Waves in Elastic Crystals, Oxford University Press, Oxford, 1999. · Zbl 0943.74002
[34] S. Oh; A. Stefanov, Improved local well-posedness for the periodic “good” Boussinesq equation, J. Differential Equations, 254, 4047-4065 (2013) · Zbl 1290.35206 · doi:10.1016/j.jde.2013.02.006
[35] A. K. Pani; H. Saranga, Finite element Galerkin method for the “good” Boussinesq equation, Nonlinear Anal., 29, 937-956 (1997) · Zbl 0880.35097 · doi:10.1016/S0362-546X(96)00093-4
[36] R. L. Sachs, On the blow-up of certain solutions of the “good” Boussinesq equation, Appl. Anal., 36, 145-152 (1990) · Zbl 0674.35082 · doi:10.1080/00036819008839928
[37] L. Tartar, Interpolation non linéairé et régularité, J. Funct. Anal., 9, 469-489 (1972) · Zbl 0241.46035 · doi:10.1016/0022-1236(72)90022-5
[38] M. Tsutsumi; T. Matahashi, On the Cauchy problem for the Boussinesq type equation, Math. Japon., 36, 371-379 (1991) · Zbl 0734.35082
[39] H. Wang; A. Esfahani, Well-posedness for the Cauchy problem associated to a periodic Boussinesq equation, Nonlinear Anal., 89, 267-275 (2013) · Zbl 1284.35353 · doi:10.1016/j.na.2013.04.011
[40] R. Xue, Local and global existence of solutions for the Cauchy problem of a generalized Boussinesq equation, J. Math. Anal. Appl., 316, 307-327 (2006) · Zbl 1106.35069 · doi:10.1016/j.jmaa.2005.04.041
[41] R. Xue, The initial-boundary value problem for the “good” Boussinesq equation on the bounded domain, J. Math. Anal. Appl., 343, 975-995 (2008) · Zbl 1143.35088 · doi:10.1016/j.jmaa.2008.02.017
[42] R. Xue, The initial-boundary-value problem for the “good” Boussinesq equation on the half line, Nonlinear Anal., 69, 647-682 (2008) · Zbl 1154.35076 · doi:10.1016/j.na.2007.06.010
[43] R. Xue, Low regularity solution of the initial-boundary-value problem for the “good” Boussinesq equation on the half line, Acta Mathematica Sinica (English Series), 26, 2421-2442 (2010) · Zbl 1210.35196 · doi:10.1007/s10114-010-7321-6
[44] Z. Yang, On local existence of solutions of initial boundary value problems for the “bad” Boussinesq-type equation, Nonlinear Anal., 51, 1259-1271 (2002) · Zbl 1022.35052 · doi:10.1016/S0362-546X(01)00894-X
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