    • Previous Article
Bifurcation diagrams and multiplicity for nonlocal elliptic equations modeling gravitating systems based on Fermi--Dirac statistics
• DCDS Home
• This Issue
• Next Article
Unilateral global bifurcation for $p$-Laplacian with non-$p-$1-linearization nonlinearity
January  2015, 35(1): 117-137. doi: 10.3934/dcds.2015.35.117

## Existence, decay and blow-up for solutions to the sixth-order generalized Boussinesq equation

 1 Laboratoire de Mathematiques Appliquées,UFRMI, Université d'Abidjan Cocody, 22 BP 582 Abidjan 22, Ivory Coast (Cote D'Ivoire)

Received  July 2010 Revised  June 2014 Published  August 2014

We study the existence, the decay and the blow-up of solutions to the Cauchy problem for the multi-dimensional generalized sixth-order Boussinesq equation: $$u_{tt} - \Delta u - \Delta^{2} u- \mu \Delta ^{3} u = \Delta f(u),\; t>0, \; x \in {\mathbb{R}^{n}}, n \geq 1,$$ where $f(u)= \gamma |u|^{p-1}u, \; \gamma \in \mathbb{R}, \; p \geq 2, \; \mu > 1/4$. We find two global existence results for appropriate initial data when $n$ verifies $1 \leq n\leq 4(p+1)/(p-1).$ On the other hand we show that if $\mu= 1/3$ and $p>13/2$, then the solution with small initial data decays in time. A blow up in finite time result is also obtained for appropriate initial data when $n$ verifies $1 \leq n\leq 4(p+1)/(p-1).$
Citation: Akmel Dé Godefroy. Existence, decay and blow-up for solutions to the sixth-order generalized Boussinesq equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 117-137. doi: 10.3934/dcds.2015.35.117
##### References:
  J. V. Boussinesq, Theorie de l'intermenscence liquide appelee onde solitaire ou de translation, se propageant dans un canal rectangulaire,, C. R. Acad. Sci. Paris, 72 (1871), 755.   Google Scholar  Y. Cho and T. Ozawa, Remarks on the Modified improved Boussinesq equations in one space dimension,, Proceeding of Royal society A, 462 (2006), 1949.  doi: 10.1098/rspa.2006.1675.  Google Scholar  F. M. Christ and M. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de-Vries equation,, Journ. funct. Anal., 100 (1991), 87.  doi: 10.1016/0022-1236(91)90103-C.  Google Scholar  C. I. Christov, G. A Maugin and M. G. Velande, Well-posed Boussinesq paradigm with purely spatial higher-order derivatives,, Phys. Rev., E54 (1996), 3621.  doi: 10.1103/PhysRevE.54.3621. Google Scholar  C. I. Christov, G. A. Maugin and A. V. Porubov, On Boussinesq's paradigm in nonlinear wave propagation,, C. R. Mecanique, 335 (2007), 521.  doi: 10.1016/j.crme.2007.08.006. Google Scholar  P. Daripa and W. Hua, A numerical method for solving an ill posed Boussinesq equation arising in water waves and nonlinear lattice,, Appl. Math. Comput., 101 (1999), 159.  doi: 10.1016/S0096-3003(98)10070-X.  Google Scholar  P. Daripa and R. K. Dash, Weakly non-local solitary wave solutions of a singularly perturbed Boussinesq equation,, Math. Comput. Simulation, 55 (2001), 393.  doi: 10.1016/S0378-4754(00)00288-3.  Google Scholar  P. Daripa, Higher-order Boussinesq equations for two-way propagation of shallow water waves,, Euro. J. Mech. B Fluids, 25 (2006), 1008.  doi: 10.1016/j.euromechflu.2006.02.003.  Google Scholar  A. Dé Godefroy, Blow up of solutions of a generalized Boussinesq equation,, IMA Journ. Math. Appl. Math, 60 (1998), 123.  doi: 10.1093/imamat/60.2.123.  Google Scholar  R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves,, Cambridge University Press, (1997).  doi: 10.1017/CBO9780511624056.  Google Scholar  O. Y. Kamenov, Exact periodic solutions of sixth-order generalized Boussinesq equation,, J. Phys. A: Math. Theor., 42 (2009).  doi: 10.1088/1751-8113/42/37/375501.  Google Scholar  T. Kato, On nonlinear Schrodinger equation II. $\mathbbH^s$ solutions and unconditional well posedness,, J. Anal. Math., 67 (1995), 281.  doi: 10.1007/BF02787794.  Google Scholar  H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_{t t}= -Au+F(u)$,, Trans. Amer. Math. Soc., 192 (1974), 1. Google Scholar  Y. Liu, Existence and Blow up of solutions of a nonlinear Pocchahammer-Chree equation,, Indianna Univ. Math. Journ., 45 (1996), 797.  doi: 10.1512/iumj.1996.45.1121.  Google Scholar  V. G. Makhankov, Dynamics of classical solutions (in non integrable systems),, Physics Reports, 35 (1978), 1.  doi: 10.1016/0370-1573(78)90074-1.  Google Scholar  G. A. Maugin, Nonlinear Waves in Elastic Crystal,, Oxford Mathematical Monographs Series, (1999). Google Scholar  R. L. Pego and M. I. Weinstein, Eigenvalues and instabilities of solitary waves,, Phil. Trans. R. Soc. Lond. Ser. A, 340 (1992), 47.  doi: 10.1098/rsta.1992.0055.  Google Scholar  M. Reed, Abstract Nonlinear Wave Equations,, Lecture Notes in Mathematics, (1976). Google Scholar  G. B. Whitham, Linear and Nonlinear Waves,, John Wiley and Sons, (1974). Google Scholar

show all references

##### References:
  J. V. Boussinesq, Theorie de l'intermenscence liquide appelee onde solitaire ou de translation, se propageant dans un canal rectangulaire,, C. R. Acad. Sci. Paris, 72 (1871), 755.   Google Scholar  Y. Cho and T. Ozawa, Remarks on the Modified improved Boussinesq equations in one space dimension,, Proceeding of Royal society A, 462 (2006), 1949.  doi: 10.1098/rspa.2006.1675.  Google Scholar  F. M. Christ and M. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de-Vries equation,, Journ. funct. Anal., 100 (1991), 87.  doi: 10.1016/0022-1236(91)90103-C.  Google Scholar  C. I. Christov, G. A Maugin and M. G. Velande, Well-posed Boussinesq paradigm with purely spatial higher-order derivatives,, Phys. Rev., E54 (1996), 3621.  doi: 10.1103/PhysRevE.54.3621. Google Scholar  C. I. Christov, G. A. Maugin and A. V. Porubov, On Boussinesq's paradigm in nonlinear wave propagation,, C. R. Mecanique, 335 (2007), 521.  doi: 10.1016/j.crme.2007.08.006. Google Scholar  P. Daripa and W. Hua, A numerical method for solving an ill posed Boussinesq equation arising in water waves and nonlinear lattice,, Appl. Math. Comput., 101 (1999), 159.  doi: 10.1016/S0096-3003(98)10070-X.  Google Scholar  P. Daripa and R. K. Dash, Weakly non-local solitary wave solutions of a singularly perturbed Boussinesq equation,, Math. Comput. Simulation, 55 (2001), 393.  doi: 10.1016/S0378-4754(00)00288-3.  Google Scholar  P. Daripa, Higher-order Boussinesq equations for two-way propagation of shallow water waves,, Euro. J. Mech. B Fluids, 25 (2006), 1008.  doi: 10.1016/j.euromechflu.2006.02.003.  Google Scholar  A. Dé Godefroy, Blow up of solutions of a generalized Boussinesq equation,, IMA Journ. Math. Appl. Math, 60 (1998), 123.  doi: 10.1093/imamat/60.2.123.  Google Scholar  R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves,, Cambridge University Press, (1997).  doi: 10.1017/CBO9780511624056.  Google Scholar  O. Y. Kamenov, Exact periodic solutions of sixth-order generalized Boussinesq equation,, J. Phys. A: Math. Theor., 42 (2009).  doi: 10.1088/1751-8113/42/37/375501.  Google Scholar  T. Kato, On nonlinear Schrodinger equation II. $\mathbbH^s$ solutions and unconditional well posedness,, J. Anal. Math., 67 (1995), 281.  doi: 10.1007/BF02787794.  Google Scholar  H. A. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_{t t}= -Au+F(u)$,, Trans. Amer. Math. Soc., 192 (1974), 1. Google Scholar  Y. Liu, Existence and Blow up of solutions of a nonlinear Pocchahammer-Chree equation,, Indianna Univ. Math. Journ., 45 (1996), 797.  doi: 10.1512/iumj.1996.45.1121.  Google Scholar  V. G. Makhankov, Dynamics of classical solutions (in non integrable systems),, Physics Reports, 35 (1978), 1.  doi: 10.1016/0370-1573(78)90074-1.  Google Scholar  G. A. Maugin, Nonlinear Waves in Elastic Crystal,, Oxford Mathematical Monographs Series, (1999). Google Scholar  R. L. Pego and M. I. Weinstein, Eigenvalues and instabilities of solitary waves,, Phil. Trans. R. Soc. Lond. Ser. A, 340 (1992), 47.  doi: 10.1098/rsta.1992.0055.  Google Scholar  M. Reed, Abstract Nonlinear Wave Equations,, Lecture Notes in Mathematics, (1976). Google Scholar  G. B. Whitham, Linear and Nonlinear Waves,, John Wiley and Sons, (1974). Google Scholar
  M. A. Christou, C. I. Christov. Fourier-Galerkin method for localized solutions of the Sixth-Order Generalized Boussinesq Equation. Conference Publications, 2001, 2001 (Special) : 121-130. doi: 10.3934/proc.2001.2001.121  Long Wei, Zhijun Qiao, Yang Wang, Shouming Zhou. Conserved quantities, global existence and blow-up for a generalized CH equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1733-1748. doi: 10.3934/dcds.2017072  Shouming Zhou, Chunlai Mu, Liangchen Wang. Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843  Pablo Álvarez-Caudevilla, Jonathan D. Evans, Victor A. Galaktionov. Gradient blow-up for a fourth-order quasilinear Boussinesq-type equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3913-3938. doi: 10.3934/dcds.2018170  Shenghao Li, Min Chen, Bing-Yu Zhang. A non-homogeneous boundary value problem of the sixth order Boussinesq equation in a quarter plane. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2505-2525. doi: 10.3934/dcds.2018104  Miao Liu, Weike Wang. Global existence and pointwise estimates of solutions for the multidimensional generalized Boussinesq-type equation. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1203-1222. doi: 10.3934/cpaa.2014.13.1203  Shuyin Wu, Joachim Escher, Zhaoyang Yin. Global existence and blow-up phenomena for a weakly dissipative Degasperis-Procesi equation. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 633-645. doi: 10.3934/dcdsb.2009.12.633  Ronghua Jiang, Jun Zhou. Blow-up and global existence of solutions to a parabolic equation associated with the fraction p-Laplacian. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1205-1226. doi: 10.3934/cpaa.2019058  Jianbo Cui, Jialin Hong, Liying Sun. On global existence and blow-up for damped stochastic nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6837-6854. doi: 10.3934/dcdsb.2019169  Xiumei Deng, Jun Zhou. Global existence and blow-up of solutions to a semilinear heat equation with singular potential and logarithmic nonlinearity. Communications on Pure & Applied Analysis, 2020, 19 (2) : 923-939. doi: 10.3934/cpaa.2020042  Xiaojing Xu. Local existence and blow-up criterion of the 2-D compressible Boussinesq equations without dissipation terms. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1333-1347. doi: 10.3934/dcds.2009.25.1333  Irena Pawłow, Wojciech M. Zajączkowski. The global solvability of a sixth order Cahn-Hilliard type equation via the Bäcklund transformation. Communications on Pure & Applied Analysis, 2014, 13 (2) : 859-880. doi: 10.3934/cpaa.2014.13.859  Barbara Kaltenbacher, Irena Lasiecka. Global existence and exponential decay rates for the Westervelt equation. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 503-523. doi: 10.3934/dcdss.2009.2.503  Guangyu Xu, Jun Zhou. Global existence and blow-up of solutions to a singular Non-Newton polytropic filtration equation with critical and supercritical initial energy. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1805-1820. doi: 10.3934/cpaa.2018086  Claudianor O. Alves, M. M. Cavalcanti, Valeria N. Domingos Cavalcanti, Mohammad A. Rammaha, Daniel Toundykov. On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 583-608. doi: 10.3934/dcdss.2009.2.583  Vo Anh Khoa, Le Thi Phuong Ngoc, Nguyen Thanh Long. Existence, blow-up and exponential decay of solutions for a porous-elastic system with damping and source terms. Evolution Equations & Control Theory, 2019, 8 (2) : 359-395. doi: 10.3934/eect.2019019  Nguyen Thanh Long, Hoang Hai Ha, Le Thi Phuong Ngoc, Nguyen Anh Triet. Existence, blow-up and exponential decay estimates for a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (1) : 455-492. doi: 10.3934/cpaa.2020023  Jun Zhou. Global existence and energy decay estimate of solutions for a class of nonlinear higher-order wave equation with general nonlinear dissipation and source term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1175-1185. doi: 10.3934/dcdss.2017064  Alejandro Sarria. Global estimates and blow-up criteria for the generalized Hunter-Saxton system. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 641-673. doi: 10.3934/dcdsb.2015.20.641  Linjie Xiong, Tao Wang, Lusheng Wang. Global existence and decay of solutions to the Fokker-Planck-Boltzmann equation. Kinetic & Related Models, 2014, 7 (1) : 169-194. doi: 10.3934/krm.2014.7.169

2018 Impact Factor: 1.143