Weakly coupled systems of semi-linear Klein-Gordon equations with memory-type dissipation

We are concerned with the existence of global in time solutions to the Cauchy problem for semi-linear Klein-Gordon equations with memory-type dissipation in $\mathbb{R}^n$. In the first place, we consider the linearized equation: applying the energy method in the Fourier space, we derive the point-wise estimate of a solution in the Fourier space. However, we found that the function in energy estimates has a mistake.