Blow Up for Klein–Gordon Equation with Nonlocal Operator and Trudinger–Moser Nonlinearity

In this paper we prove a blow up result for a class of nonlocal scalar Klein–Gordon equation. We assume that the nonlinearity has critical exponential growth. Additionally, we prove that the ground state solution of the elliptic problem associated to the original problem is unstable. The strategy is to adapt the recent ideas of Carrião et al. (J Dyn Differ Equ 2023. https://doi.org/10.1007/s10884-023-10281-3) to find two regions (stable and unstable regions). Since the one dimensional case combined with the 1/2 Laplacian operator cause lack of the control on the \(L^2(\mathbb {R})\) $L^2(\mathbb{R})$ norm of \((-\Delta )^{\frac{1}{4}}u\) ${(-\mathrm{\Delta })}^{\frac{1}{4}}u$ , new delicate calculations are necessary. We prove also that there exists a subset of \(H^{\frac{1}{2}}(\mathbb {R})\times L^2(\mathbb {R})\) $H^{\frac{1}{2}}(\mathbb{R})\times L^2(\mathbb{R})$ such that the solution is global when the initial data is taken into this set.