Sharp threshold of global existence and blow-up of the combined nonlinear Klein–Gordon equation

In this paper, we show the global well-posedness and blow-up result of the solutions with the energy below the threshold for the combined nonlinear Klein–Gordon equation u _tt −Δu+u=|u| ^p−1 u−|u| ^q−1 u,d≥3, in the energy space H ¹ (R ^d )×L ² (R ^d ), where 1+[Formula presented]<q<p≤1+[Formula presented]. We give a threshold of blow up and global well-posedness using a modified variational approach, in the spirit of Ibrahim et al. (2011) [7].