On global existence and blow up of weak solutions for a wave equation with mixed local and nonlocal propagation

In this paper, we investigate the initial boundary value problem of a wave equation with mixed local and nonlocal propagation. First, we introduce the spatial framework to study the wave equation, which is the intersection of a classical Sobolev space and a fractional Sobolev space. By the Mountain Pass Theorem, we obtain the attainability of the optimal embedding constant from the introduced space to the suitable Lebesgue space. Second, by introducing a family of potential wells in the introduced space, we obtain the existence of a global weak solution through the utilization of potential well theory. At last, by analyzing the properties of the energy functional, we show that any weak solution must blow up at the existence time.