Asymptotic profile of solutions for strongly damped Klein-Gordon equations

TL;DR

This paper analyzes the long-term behavior of solutions to strongly damped Klein-Gordon equations, revealing that the wave effect diminishes due to the mass term, especially in low dimensions, using a simple asymptotic analysis method.

Contribution

It provides the first detailed asymptotic profiles for solutions of strongly damped Klein-Gordon equations with weighted initial data, highlighting the impact of the mass term in low dimensions.

Findings

Wave effect is weak due to the mass term in low dimensions

Asymptotic profiles are derived using Ikehata's method

Results show differences from strongly damped wave equations without mass

Abstract

We consider the Cauchy problem in the whole space for strongly damped Klein-Gordon equations. We derive asymptotic profles of solutions with weighted initial data by a simple method introduced by R. Ikehata. The obtained results show that the wave effect will be weak because of the mass term, especially in the low dimensional case (n = 1,2) as compared with the strongly damped wave equations without mass term (m = 0), so the most interesting topic in this paper is the n = 1,2 cases.