Remarks on decay effects of regularity loss type wave equations with structural damping terms

TL;DR

This paper investigates the decay behavior of solutions to a class of wave equations with structural damping, providing new results that confirm the solutions' energy actually decays over time, unlike previous boundedness results.

Contribution

It establishes the decay of total energy for regularity loss type wave equations with structural damping, resolving an open question from prior research.

Findings

Proves energy decay for solutions with initial data in the energy space

Extends decay estimates to a broader class of wave equations with damping

Provides a positive answer to the decay question left open in previous studies

Abstract

After GGH model was proposed by M. Ghisi, M. Gobbino and A. Haraux (2016), R. Ikehata and S. Iyota (2018) showed decay estimates for the total energy of solutions to GGH equations uniformly in the initial data. However, their results imply that the total energy is bounded when the initial data belong to the energy space. That is, whether it actually decays has not been known so far. In this paper we report a positive answer to that question.