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) - Delta u + u = vertical bar u vertical bar(p-1)u - vertical bar u vertical bar(q-1)u, d >= 3, in the energy space H-1(R-d) x L-2 (R-d), where 1 + 4/d < q < p <= 1 + 4/d-2. 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). (C) 2019 Elsevier Inc. All rights reserved.