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

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 $-\mathrm{\mid }\mathrm{\mid }\mathrm{\nabla }{u}_t(t)\mathrm{\mid }{\mathrm{\mid }}_2^2\mathrm{\Delta }{u}_t$. 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 $1\mathrm{/}t$, 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.