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

TL;DR

This paper investigates the existence of global solutions for semi-linear Klein-Gordon equations with memory-type dissipation, using Fourier analysis and energy methods, while identifying a key mistake in the energy estimates.

Contribution

It provides a detailed Fourier space analysis of the linearized equation and highlights an error in the energy estimate approach for such dissipative systems.

Findings

Derived point-wise Fourier estimates for the linearized equation

Identified a mistake in the energy estimate function

Contributed to the understanding of dissipative Klein-Gordon equations

Abstract

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.