American Institute of Mathematical Sciences

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May  2018, 38(5): 2505-2525. doi: 10.3934/dcds.2018104

A non-homogeneous boundary value problem of the sixth order Boussinesq equation in a quarter plane

Shenghao Li 1, , Min Chen 1, and  Bing-Yu Zhang 2,3,,

1. 

Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA
2. 

Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025, USA
3. 

Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, China

*Corresponding author: Bing-Yu Zhang

Received  November 2016 Revised  December 2017 Published  March 2018

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:
$\begin{equation}\label{0}\begin{cases}u_{tt}-u_{xx}+β u_{xxxx}-u_{xxxxxx}+(u^2)_{xx} = 0, \, \, \, \, \,\,\,\,\, \mbox{for }x>0\mbox{, }t>0, \\u(x, 0) = \varphi (x), u_t(x, 0) = ψ "(x), \\u(0, t) = h_1(t), u_{xx}(0, t) = h_2(t), u_{xxxx}(0, t) = h_3(t), \end{cases}\, \, \, \, \, \, \, \, \, \, (1)\end{equation}$
where
$β = ± 1$
 . It is shown that the problem is locally well-posed in the space $H^s(\mathbb{R}^+)$ for any 0≤s<
$\frac{13}{2}$
 with the initial data
$ (\varphi, ψ)$
 in the space
$H^s(\mathbb{R}^+)× H^{s-1}(\mathbb{R}^+)$
and the naturally compatible boundary data
$\mbox{ $h_1∈ H_{loc}^{\frac{s+1}{3}}(\mathbb{R}^+)$, $h_2∈ H_{loc}^{\frac{s-1}{3}}(\mathbb{R}^+) \text{and}\,\,\, h_3∈ H_{loc}^{\frac{s-3}{3}}(\mathbb{R}^+)$}$
with optimal regularity.
Keywords: Sixth order Boussinesq equation, nonhomogeneous boundary value problem, well-posedness, boundary integral operators.
Mathematics Subject Classification: Primary: 35Q53, 35Q55; Secondary: 35Q35.
Citation: Shenghao Li, Min Chen, Bing-Yu Zhang. A non-homogeneous boundary value problem of the sixth order Boussinesq equation in a quarter plane. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2505-2525. doi: 10.3934/dcds.2018104
References:
[1]

J. Bergh and J. Lofstrom, Interpolation Spaces: An Introduction, Springer-Verlag Berlin Heidelberg, New York, 1976.

[2]

J. L. Bona and M. Chen, A boussinesq system for two-way propagation of nonlinear dispersive waves, Physica D: Nonlinear Phenomena, 116 (1998), 191-224.  doi: 10.1016/S0167-2789(97)00249-2.

[3]

J. L. Bona, M. Chen and J. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. ii. the nonlinear theory, Nonlinearity, 17 (2004), 925-952.  doi: 10.1088/0951-7715/17/3/010.

[4]

J. L. Bona, M. Chen and 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 (2002), 283-318.  doi: 10.1007/s00332-002-0466-4.

[5]

J. L. Bona and R. L. Sachs, Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation, Comm. Math. Phys., 118 (1998), 15-29.  doi: 10.1007/BF01218475.

[6]

J. L. Bona, S. M. Sun and B.-Y. Zhang, A non-homogeneous boundary-value problem for the Korteweg-de Vries equation in a quarter plane, Trans. Amer, Math. Soc., 354 (2002), 427-490.  doi: 10.1090/S0002-9947-01-02885-9.

[7]

J. L. Bona, S. M. Sun and B.-Y. Zhang, A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain, Comm. Partial Differential Equations, 28 (2003), 1391-1436.  doi: 10.1081/PDE-120024373.

[8]

J. L. Bona, S. M. Sun and 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 (2009), 2558-2596.  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]. doi: 10.1016/j.matpur.2017.11.001.

[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 (1872), 55-108. 

[11]

C. Christov, G. Maugin and M. Velarde, Well-posed Boussinesq paradigm with purely spatial higher-order derivatives, Phys. Rev. E, 54 (1996), 3621-3638.  doi: 10.1103/PhysRevE.54.3621.

[12]

J. E. Colliander and C. E. Kenig, The generalized Korteweg-de Vries equation on the half line, Comm. Partial Differential Equations, 27 (2002), 2187-2266.  doi: 10.1081/PDE-120016157.

[13]

J. de Frutos, T. Ortega and J. M. Sanz-Serna, Pseudospectral method for the "good" Boussinesq equation, Math. Comp., 57 (1991), 109-122. 

[14]

A. Esfahani and L. G. Farah, Local well-posedness for the sixth-order Boussinesq equation, Journal of Mathematical Analysis and Applications, 385 (2012), 230-242.  doi: 10.1016/j.jmaa.2011.06.038.

[15]

A. Esfahani, L. G. Farah and H. Wang, Global existence and blow-up for the generalized sixth-order Boussinesq equation, Nonlinear Anal., 75 (2012), 4325-4338.  doi: 10.1016/j.na.2012.03.019.

[16]

A. Esfahani and H. Wang, A bilinear estimate with application to the sixth-order Boussinesq equation, Differential Integral Equations, 27 (2014), 401-414. 

[17]

Y.-F. Fang and M. G. Grillakis, Existence and uniqueness for Boussinesq type equations on a circle, Comm.Partial Differential Equations, 21 (1996), 1253-1277.  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 (2009), 52-73.  doi: 10.1080/03605300802682283.

[19]

L. G. Farah and M. Scialom, On the periodic "good " Boussinesq equation, Proc. Amer. Math. Soc., 138 (2010), 953-964.  doi: 10.1090/S0002-9939-09-10142-9.

[20]

B.-F. Feng, T. Kawahara, T. Mitsui and Y.-S. Chan, Solitary-wave propagation and interactions for a sixth-order generalized Boussinesq equation, Int. J. Math. Math. Sci., 2005 (2005), 1435-1448. 

[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 (2005), 647-668. 

[22]

R. Hunt, Muckenhoupt, W. Benjamin and R. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc., 176 (1973), 227-251.  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.

[24]

C.E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620.  doi: 10.1002/cpa.3160460405.

[25]

C.E. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. l Soc., 9 (1996), 573-603.  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 (1993), 257-293.  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.

[28]

F.-L. Liu and D. L. Russell, Solutions of the Boussinesq equation on a periodic domain, J. Math. Anal. Appl., 194 (1995), 78-102.  doi: 10.1006/jmaa.1995.1287.

[29]

Y. Liu, Instability of solitary waves for generalized Boussinesq equations, J. Dynam. Differential Equations, 5 (1993), 537-558.  doi: 10.1007/BF01053535.

[30]

Y. Liu, Instability and blow-up of solutions to a generalized Boussinesq equation, SIAM J. Math. Anal., 26 (1995), 1527-1546.  doi: 10.1137/S0036141093258094.

[31]

Y. Liu, Decay and scattering of small solutions of a generalized Boussinesq equation, J. Funct. Anal., 147 (1997), 51-68.  doi: 10.1006/jfan.1996.3052.

[32]

Y. Liu, Strong instability of solitary-wave solutions of a generalized Boussinesq equation, J. Differential Equations, 164 (2000), 223-239.  doi: 10.1006/jdeq.2000.3765.

[33]

G. A. Maugin, Nonlinear Waves in Elastic Crystals, Oxford University Press, Oxford, 1999.

[34]

S. Oh and A. Stefanov, Improved local well-posedness for the periodic "good" Boussinesq equation, J. Differential Equations, 254 (2013), 4047-4065.  doi: 10.1016/j.jde.2013.02.006.

[35]

A.K. Pani and H. Saranga, Finite element Galerkin method for the "good" Boussinesq equation, Nonlinear Anal., 29 (1997), 937-956.  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 (1990), 145-152.  doi: 10.1080/00036819008839928.

[37]

L. Tartar, Interpolation non linéairé et régularité, J. Funct. Anal., 9 (1972), 469-489.  doi: 10.1016/0022-1236(72)90022-5.

[38]

M. Tsutsumi and T. Matahashi, On the Cauchy problem for the Boussinesq type equation, Math. Japon., 36 (1991), 371-379. 

[39]

H. Wang and A. Esfahani, Well-posedness for the Cauchy problem associated to a periodic Boussinesq equation, Nonlinear Anal., 89 (2013), 267-275.  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 (2006), 307-327.  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 (2008), 975-995.  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 (2008), 647-682.  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 (2010), 2421-2442.  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 (2002), 1259-1271.  doi: 10.1016/S0362-546X(01)00894-X.

show all references

References:
[1]

J. Bergh and J. Lofstrom, Interpolation Spaces: An Introduction, Springer-Verlag Berlin Heidelberg, New York, 1976.

[2]

J. L. Bona and M. Chen, A boussinesq system for two-way propagation of nonlinear dispersive waves, Physica D: Nonlinear Phenomena, 116 (1998), 191-224.  doi: 10.1016/S0167-2789(97)00249-2.

[3]

J. L. Bona, M. Chen and J. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. ii. the nonlinear theory, Nonlinearity, 17 (2004), 925-952.  doi: 10.1088/0951-7715/17/3/010.

[4]

J. L. Bona, M. Chen and 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 (2002), 283-318.  doi: 10.1007/s00332-002-0466-4.

[5]

J. L. Bona and R. L. Sachs, Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation, Comm. Math. Phys., 118 (1998), 15-29.  doi: 10.1007/BF01218475.

[6]

J. L. Bona, S. M. Sun and B.-Y. Zhang, A non-homogeneous boundary-value problem for the Korteweg-de Vries equation in a quarter plane, Trans. Amer, Math. Soc., 354 (2002), 427-490.  doi: 10.1090/S0002-9947-01-02885-9.

[7]

J. L. Bona, S. M. Sun and B.-Y. Zhang, A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain, Comm. Partial Differential Equations, 28 (2003), 1391-1436.  doi: 10.1081/PDE-120024373.

[8]

J. L. Bona, S. M. Sun and 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 (2009), 2558-2596.  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]. doi: 10.1016/j.matpur.2017.11.001.

[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 (1872), 55-108. 

[11]

C. Christov, G. Maugin and M. Velarde, Well-posed Boussinesq paradigm with purely spatial higher-order derivatives, Phys. Rev. E, 54 (1996), 3621-3638.  doi: 10.1103/PhysRevE.54.3621.

[12]

J. E. Colliander and C. E. Kenig, The generalized Korteweg-de Vries equation on the half line, Comm. Partial Differential Equations, 27 (2002), 2187-2266.  doi: 10.1081/PDE-120016157.

[13]

J. de Frutos, T. Ortega and J. M. Sanz-Serna, Pseudospectral method for the "good" Boussinesq equation, Math. Comp., 57 (1991), 109-122. 

[14]

A. Esfahani and L. G. Farah, Local well-posedness for the sixth-order Boussinesq equation, Journal of Mathematical Analysis and Applications, 385 (2012), 230-242.  doi: 10.1016/j.jmaa.2011.06.038.

[15]

A. Esfahani, L. G. Farah and H. Wang, Global existence and blow-up for the generalized sixth-order Boussinesq equation, Nonlinear Anal., 75 (2012), 4325-4338.  doi: 10.1016/j.na.2012.03.019.

[16]

A. Esfahani and H. Wang, A bilinear estimate with application to the sixth-order Boussinesq equation, Differential Integral Equations, 27 (2014), 401-414. 

[17]

Y.-F. Fang and M. G. Grillakis, Existence and uniqueness for Boussinesq type equations on a circle, Comm.Partial Differential Equations, 21 (1996), 1253-1277.  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 (2009), 52-73.  doi: 10.1080/03605300802682283.

[19]

L. G. Farah and M. Scialom, On the periodic "good " Boussinesq equation, Proc. Amer. Math. Soc., 138 (2010), 953-964.  doi: 10.1090/S0002-9939-09-10142-9.

[20]

B.-F. Feng, T. Kawahara, T. Mitsui and Y.-S. Chan, Solitary-wave propagation and interactions for a sixth-order generalized Boussinesq equation, Int. J. Math. Math. Sci., 2005 (2005), 1435-1448. 

[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 (2005), 647-668. 

[22]

R. Hunt, Muckenhoupt, W. Benjamin and R. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc., 176 (1973), 227-251.  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.

[24]

C.E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620.  doi: 10.1002/cpa.3160460405.

[25]

C.E. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. l Soc., 9 (1996), 573-603.  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 (1993), 257-293.  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.

[28]

F.-L. Liu and D. L. Russell, Solutions of the Boussinesq equation on a periodic domain, J. Math. Anal. Appl., 194 (1995), 78-102.  doi: 10.1006/jmaa.1995.1287.

[29]

Y. Liu, Instability of solitary waves for generalized Boussinesq equations, J. Dynam. Differential Equations, 5 (1993), 537-558.  doi: 10.1007/BF01053535.

[30]

Y. Liu, Instability and blow-up of solutions to a generalized Boussinesq equation, SIAM J. Math. Anal., 26 (1995), 1527-1546.  doi: 10.1137/S0036141093258094.

[31]

Y. Liu, Decay and scattering of small solutions of a generalized Boussinesq equation, J. Funct. Anal., 147 (1997), 51-68.  doi: 10.1006/jfan.1996.3052.

[32]

Y. Liu, Strong instability of solitary-wave solutions of a generalized Boussinesq equation, J. Differential Equations, 164 (2000), 223-239.  doi: 10.1006/jdeq.2000.3765.

[33]

G. A. Maugin, Nonlinear Waves in Elastic Crystals, Oxford University Press, Oxford, 1999.

[34]

S. Oh and A. Stefanov, Improved local well-posedness for the periodic "good" Boussinesq equation, J. Differential Equations, 254 (2013), 4047-4065.  doi: 10.1016/j.jde.2013.02.006.

[35]

A.K. Pani and H. Saranga, Finite element Galerkin method for the "good" Boussinesq equation, Nonlinear Anal., 29 (1997), 937-956.  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 (1990), 145-152.  doi: 10.1080/00036819008839928.

[37]

L. Tartar, Interpolation non linéairé et régularité, J. Funct. Anal., 9 (1972), 469-489.  doi: 10.1016/0022-1236(72)90022-5.

[38]

M. Tsutsumi and T. Matahashi, On the Cauchy problem for the Boussinesq type equation, Math. Japon., 36 (1991), 371-379. 

[39]

H. Wang and A. Esfahani, Well-posedness for the Cauchy problem associated to a periodic Boussinesq equation, Nonlinear Anal., 89 (2013), 267-275.  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 (2006), 307-327.  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 (2008), 975-995.  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 (2008), 647-682.  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 (2010), 2421-2442.  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 (2002), 1259-1271.  doi: 10.1016/S0362-546X(01)00894-X.

Figure .  Sketch of the half line case
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Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. I. Well-posedness and convergence of the method of lines. Inverse Problems and Imaging, 2013, 7 (2) : 307-340. doi: 10.3934/ipi.2013.7.307

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