Blow-up phenomena for the sixth-order Boussinesq equation with fourth-order dispersion term and nonlinear source

This paper deals with the sixth-order Boussinesq equation with fourth-order dispersion term and nonlinear source. By using some ordinary differential inequalities, the conditions on finite time blow-up of solutions are given with suitable assumptions on initial values. Moreover, the upper and lower bounds of the blow-up time are also investigated.

[1]  Runzhang Xu,et al.  Low regularity of solutions to the Rotation-Camassa-Holm type equation with the Coriolis effect , 2020, Discrete & Continuous Dynamical Systems - A.

[2]  C. Christov,et al.  Well-posed Boussinesq paradigm with purely spatial higher-order derivatives. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[3]  D. Sattinger,et al.  Saddle points and instability of nonlinear hyperbolic equations , 1975 .

[4]  J. Boussinesq,et al.  Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. , 1872 .

[5]  Runzhang Xu.  Cauchy problem of generalized Boussinesq equation with combined power‐type nonlinearities , 2011 .

[6]  L. Farah,et al.  Local well-posedness for the sixth-order Boussinesq equation , 2010, 1012.5088.

[7]  A. D. Godefroy.  Existence, decay and blow-up for solutions to the sixth-order generalized Boussinesq equation , 2014 .

[8]  Runzhang Xu,et al.  Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation , 2020, Advances in Nonlinear Analysis.

[9]  Jun Zhou,et al.  Global existence and blow-up of solutions for a Kirchhoff type plate equation with damping , 2015, Appl. Math. Comput..

[10]  Q. Lin,et al.  On the Cauchy problem for a generalized Boussinesq equation , 2009 .

[11]  Jorge A. Esquivel-avil.  Blow-up in damped abstract nonlinear equations , 2020, Electronic Research Archive.

[12]  Jun Zhou,et al.  Well-Posedness of Solutions for the Sixth-Order Boussinesq Equation with Linear Strong Damping and Nonlinear Source , 2021, Journal of Nonlinear Science.

[13]  Howard A. Levine,et al.  Some Additional Remarks on the Nonexistence of Global Solutions to Nonlinear Wave Equations , 1974 .

[14]  Gongwei Liu.  The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term , 2020, Electronic Research Archive.

[15]  F. Linares.  Global Existence of Small Solutions for a Generalized Boussinesq Equation , 1993 .

[16]  Howard A. Levine,et al.  Instability and Nonexistence of Global Solutions to Nonlinear Wave Equations , 1974 .

[17]  Xiaoyan Su,et al.  The initial-boundary value problem for the generalized double dispersion equation , 2017, Zeitschrift für angewandte Mathematik und Physik.

[18]  Runzhang Xu,et al.  Global existence and blow up of solutions for Cauchy problem of generalized Boussinesq equation , 2008 .

[19]  Randall J. LeVeque,et al.  Solitary‐Wave Interactions in Elastic Rods , 1986 .

[20]  Prabir Daripa,et al.  Higher-order Boussinesq equations for two-way propagation of shallow water waves , 2006 .

[21]  Huan Zhang,et al.  Asymptotic behaviors of solutions to a sixth-order Boussinesq equation with logarithmic nonlinearity , 2021, Communications on Pure & Applied Analysis.

[22]  Ru-ying Xue.  Local and global existence of solutions for the Cauchy problem of a generalized Boussinesq equation , 2006 .

[23]  Meng-Rong Li,et al.  Existence and nonexistence of global solutions of some system of semilinear wave equations , 2003 .

[24]  Huiyang Xu,et al.  Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity , 2019, Discrete & Continuous Dynamical Systems - A.

[25]  Alexey V. Porubov,et al.  On Boussinesq s paradigm in nonlinear wave propagation , 2007 .

[26]  Yue Liu,et al.  Instability and blow-up of solutions to a generalized Boussinesq equation , 1995 .

[27]  J. Bona,et al.  Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation , 1988 .

[28]  Wei Hua,et al.  A numerical study of an ill-posed Boussinesq equation arising in water waves and nonlinear lattices: Filtering and regularization techniques , 1999, Appl. Math. Comput..

[29]  W. Lian,et al.  Global existence and blow up of solutions for pseudo-parabolic equation with singular potential , 2020, Journal of Differential Equations.

[30]  Jun Zhou,et al.  Initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity , 2020, Electronic Research Archive.

[31]  Xu Runzhang,et al.  Global well-posedness of coupled parabolic systems , 2020 .

[32]  S. Deng.  Generalized multi-hump wave solutions of Kdv-Kdv system of Boussinesq equations , 2019, Discrete & Continuous Dynamical Systems - A.

[33]  Ji-hong Shen,et al.  The initial-boundary value problems for a class of sixth order nonlinear wave equation , 2017 .

[34]  Qiang Liu,et al.  Global existence and blow-up of weak solutions for a class of fractional p-Laplacian evolution equations , 2020, Advances in Nonlinear Analysis.

[35]  Shubin Wang,et al.  The Cauchy Problem for the Generalized IMBq Equation in Ws, p(Rn) , 2002 .

[36]  V. G. Makhankov,et al.  Dynamics of classical solitons (in non-integrable systems) , 1978 .

[37]  W. Lian,et al.  Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term , 2019, Advances in Nonlinear Analysis.