The purpose of this paper is to study the following equation driven by a nonlocal integro‐differential operator LK$$ {\mathcal{L}}_K $$: utt+[u]s2(θ−1)LKu+a|ut|m−1ut=b|u|p−1u,$$ {u}_{tt}+{\left[u\right]}_s^{2\left(\theta -1\right)}{\mathcal{L}}_Ku+a{\left|{u}_t\right|}^{m-1}{u}_t=b{\left|u\right|}^{p-1}u, $$with homogeneous Dirichlet boundary condition and initial data, where [u]s2$$ {\left[u\right]}_s^2 $$ is the Gagliardo seminorm, a≥0,b>0,0