Asymptotic behavior of the wave equation subject to a Kelvin-Voigt nonlocal damping

TL;DR

This paper analyzes the wave equation with Kelvin-Voigt nonlocal damping, establishing solution existence and showing that energy decay follows an optimal 1/t rate, advancing understanding of such dissipative systems.

Contribution

It demonstrates the existence of strong and weak solutions for the wave equation with nonlocal damping and characterizes the optimal decay rate of energy.

Findings

Solutions exist both strongly and weakly.

Energy decay rate is optimally 1/t.

The damping structure effectively dissipates energy.

Abstract

In this article, we examine the well-posedness and asymptotic behavior of the energy associated with the wave equation that incorporates a Kelvin-Voigt nonlocal damping structure given by $-||\nabla u_t(t)||_2^2 \Delta u_t$. Utilizing the robust framework of nonlinear semigroups, we successfully demonstrate the existence of both strong and weak solutions. Our findings reveal that the decay rate for these solutions is optimally characterized by $1/t$, highlighting the effectiveness of this dissipative structure. This work not only enhances our understanding of the wave equation under nonlocal damping but also emphasizes the crucial balance between mathematical rigor and physical relevance.