Based on the possible singularity of stationary state, we revisit the initial boundary value problem of the classical Klein-Gordon-Schrödinger (KGS) system in one space dimension. The wellposedness is established in a class of Sobolev NLS solutions together with exponentially growing KG solutions.

In this paper, we investigate the finite-time blowup phenomenon in the fractional Klein-Gordon-Schrödinger (FKGS) system featuring nongauge-invariant power-type nonlinearities on ${\mathbb{R}}^n$. The system models interactions between nucleon and meson fields, augmented by fractional Laplacian operators and nonlinear terms that disrupt conservation laws. By introducing a novel test function tailored to address the difficulties posed by mixed fractional operators and the absence of energy conservation, we established sufficient conditions for finite-time blowup under specific initial data constraints. Our analysis revealed that solutions fail to exist globally when the nonlinear exponents and initial energy satisfy critical inequalities, with the lifespan bounded by a power-law dependence on the problem parameters.