<p>This article delves into the study of elastic structures with viscoelastic boundary conditions, focusing on an elastic thin plate in a bounded domain Ω ⊂ ⊂² with ⊂²- smooth boundary Γ. The plate is clamped, and memory effects are considered on a subset Γ₀ with positive boundary measure. The vertical deflection ⊂(⊂, ⊂) of this thin elastic plate is governed by a partial differential equation involving wave equations and memory effects. The specific problem can be described by the following equations: ⊂⊂⊂(⊂, ⊂) + Δ²⊂(⊂, ⊂) = 0, in Ω × ⊂⁺, (1.1a) ⊂(⊂, ⊂) = ∂⊂⊂(⊂, ⊂) = 0, on Γ₀ × ⊂⁺, (1.1b) ⊂₁⊂(⊂, ⊂) − ∫₀⁺∞ ⊂′(⊂) ∂⊂[⊂(⊂, ⊂) − ⊂(⊂, ⊂ − ⊂)]d⊂ = 0, on Γ₀ × ⊂⁺, (1.1c) ⊂₂⊂(⊂, ⊂) + ∫₀⁺∞ ⊂′(⊂)[⊂(⊂, ⊂) − ⊂(⊂, ⊂ − ⊂)]d⊂ = ⊂(⊂, ⊂), on Γ₁ × ⊂⁺, (1.1d) ⊂(⊂, 0⁺) = ⊂₀(⊂), ⊂⊂(⊂, 0⁺) = ⊂₁(⊂), (1.1e) ⊂(⊂, −⊂) = ⊂(⊂, ⊂), for 0 &lt; ⊂ &lt; ∞. (1.1f) This study explores the interplay between elastic structures and viscoelastic boundary conditions, examining the behavior of the thin plate under these specified conditions.</p>