Energy asymptotics for the strongly damped Klein-Gordon equation

TL;DR

This paper investigates the long-term energy behavior of solutions to the strongly damped Klein-Gordon equation, showing exponential decay for zero mean solutions and analyzing energy limits for small initial data.

Contribution

It provides new insights into the asymptotic energy behavior of solutions, including conditions for decay and non-zero limits, for the strongly damped Klein-Gordon equation.

Findings

Exponential decay of energy for zero mean solutions.

Characterization of energy limits for small initial data.

Proof that energy limit need not be zero.

Abstract

We consider the strongly damped Klein Gordon equation for defocusing nonlinearity and we study the asymptotic behaviour of the energy for periodic solutions. We prove first the exponential decay to zero for zero mean solutions. Then, we characterize the limit of the energy, when the time tends to infinity, for solutions with small enough initial data and we finally prove that such limit is not necessary zero.