Global solutions and blow-up for Klein–Gordon equation with damping and logarithmic terms

Abstract

In this paper, the initial boundary value problem for Klein–Gordon equation with weak and strong damping terms and nonlinear logarithmic term is investigated, which is known as one of the nonlinear wave equations in relativistic quantum mechanics and quantum field theory. Firstly, we prove the local existence and uniqueness of weak solution by using the Galerkin method and Contraction mapping principle. The global existence, energy decay and finite time blow-up of the solution with subcritical initial energy are established. Then these conclusions are extended to the critical initial energy. Besides, the finite time blow-up result with supercritical initial energy is shown.

KEYWORDS:

• Klein–Gordon equation
• weak and strong damping terms
• logarithmic nonlinear term
• global solution

AMS SUBJECT CLASSIFICATION:

• 35-02

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant number 11271141].