Sign-preserving functionals and blow-up to Klein–Gordon equation with arbitrary high energy

Abstract

The global behaviour of the weak solutions of the Cauchy problem to nonlinear Klein–Gordon equation with combined power-type nonlinearity is studied. Finite time blow-up of the solutions with arbitrary high positive initial energy is proved under general structural conditions on the initial data. A new functional, invariant under the flow of the equation, is introduced and investigated.

Keywords:

• Klein–Gordon equation
• combined power-type nonlinearity
• blow-up
• arbitrary high positive initial energy
• sign-preserving functionals

AMS Subject Classifications:

• 35L15
• 35L05
• 35B44

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was partially supported by the Bulgarian Science Fund under grant DFNI I-02/9.