In this paper, we conduct a comprehensive study of the global well-posedness of solution for a class of nonlocal wave equations with variable-order fractional Laplacian and variable exponent nonlinearity by constructing a suitable framework of the variational theory. We first prove the local-in-time existence of the weak solution via the Galerkin approximation technique and fixed point theory. Then by constructing the potential well theory, we classify the initial data leading to the global existence and finite time blowup of the solution for three different initial energy cases, that is, subc...
