Global behavior of the solutions to Boussinesq type equation with linear restoring force

Nonexistence of global solutions to new ordinary differential inequality and applications to nonlineardispersive equations

“…By means of the properties of these functionals, the proof of blow up result follows from Lemma (i). An application of this method is given in for Klein–Gordon equation and in for generalized Boussinesq equation. The second approach uses directly the blow up results of Lemma ; see for Klein–Gordon equation and for generalized Boussinesq equation.Remark For α = 0, Theorem (ii) coincides with the result in Lemma , and the corresponding upper bounds of the blow up time are one and the same, that is, .…”

“…Let us recall that in case of subcritical and critical initial energy (0 < E (0)≤ d ), the global behaviour of the weak solutions is fully characterized by means of the potential well method, for example, – for Klein–Gordon equation – and – for generalized Boussinesq equation – with combined power‐type and exponential‐type nonlinearities. Here, d is the depth of the potential well, defined in , while the initial energy E (0) is given by and , respectively.…”

“…Let us recall that for generalized Boussinesq equation, –, with m = 0, nonexistence of global solutions with arbitrary high positive energy is proven in for nonlinearities f ( u ), satisfying . For m > 0, the first blow up result for the solutions with arbitrary high positive energy is obtained in for problem – and nonlinear term . In all these papers , the proof is based on sign invariant functionals.…”

“…For m > 0, the first blow up result for the solutions with arbitrary high positive energy is obtained in for problem – and nonlinear term . In all these papers , the proof is based on sign invariant functionals. Note that in the present paper, we study problem – with very general nonlinearities f ( u ) satisfying either ( H 1) or ( H 2), while in , a spacial case of ( H 1) is only considered.…”

“…In all these papers , the proof is based on sign invariant functionals. Note that in the present paper, we study problem – with very general nonlinearities f ( u ) satisfying either ( H 1) or ( H 2), while in , a spacial case of ( H 1) is only considered. Moreover, the result in Section 4.2 essentially improves the blow up result from .…”