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A non-homogeneous boundary value problem of the sixth order Boussinesq equation in a quarter plane
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 |
$\begin{equation}\label{0}\begin{cases}u_{tt}-u_{xx}+β u_{xxxx}-u_{xxxxxx}+(u^2)_{xx} = 0, \, \, \, \, \,\,\,\,\, \mbox{for }x>0\mbox{, }t>0, \\u(x, 0) = \varphi (x), u_t(x, 0) = ψ "(x), \\u(0, t) = h_1(t), u_{xx}(0, t) = h_2(t), u_{xxxx}(0, t) = h_3(t), \end{cases}\, \, \, \, \, \, \, \, \, \, (1)\end{equation}$ |
$β = ± 1$ |
$\frac{13}{2}$ |
$ (\varphi, ψ)$ |
$H^s(\mathbb{R}^+)× H^{s-1}(\mathbb{R}^+)$ |
$\mbox{ $h_1∈ H_{loc}^{\frac{s+1}{3}}(\mathbb{R}^+)$, $h_2∈ H_{loc}^{\frac{s-1}{3}}(\mathbb{R}^+) \text{and}\,\,\, h_3∈ H_{loc}^{\frac{s-3}{3}}(\mathbb{R}^+)$}$ |
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. |

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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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