Global existence of Cauchy problem for Boussinesq paradigm equation

In this paper we study the Cauchy problem for the Boussinesq paradigm equation u"t"[email protected]@b"[email protected]"t"[email protected]"[email protected]^[email protected](u), where f(u)[email protected]|u|^p. New functionals are introduced and their sign preserving properties under the flow of the equation are studied. The existence of a global weak solution with supercritical initial energy is proved. The explicit value of the critical energy constant is found.

[1]  Runzhang Xu,et al.  Global existence of solution for Cauchy problem of multidimensional generalized double dispersion equations , 2009 .

[2]  O. Ladyzhenskaya,et al.  The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types , 1978 .

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

[4]  C. Christov,et al.  Theoretical and Numerical Aspects for Global Existence and Blow Up for the Solutions to Boussinesq Paradigm Equation , 2011 .

[5]  Christo I. Christov,et al.  An energy-consistent dispersive shallow-water model , 2001 .

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

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

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

[9]  Elliott H. Lieb,et al.  Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities , 1983 .

[10]  S. Turitsyn,et al.  Nonstable solitons and sharp criteria for wave collapse. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[11]  Shubin Wang,et al.  Cauchy problem of the generalized double dispersion equation , 2006 .

[12]  Runzhang Xu,et al.  Finite time Blow up of solution of Cauchy problem for the multidimensional generalized double dispersion equations with arbitrary initial energy , 2011, 2011 International Conference on Multimedia Technology.

[13]  Turitsyn.  Blow-up in the Boussinesq equation. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[14]  N. Polat,et al.  Existence and blow-up of solution of Cauchy problem for the generalized damped multidimensional Boussinesq equation , 2009 .

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

[16]  Runzhang Xu,et al.  Global existence and nonexistence of solution for Cauchy problem of multidimensional double dispersion equations , 2009 .

[17]  Milena Dimova,et al.  Comparison of Some Finite Difference Schemes for Boussinesq Paradigm Equation , 2011, MMCP.

[18]  N. Kolkovska,et al.  A new conservative finite difference scheme for Boussinesq paradigm equation , 2012 .

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

[20]  M. Kwong.  Uniqueness of positive solutions of Δu−u+up=0 in Rn , 1989 .

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