Abstract

<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 ⊂ ℝ² 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 × ℝ⁺, (1.1a) ????(????, ????) = ∂????????(????, ????) = 0, on Γ₀ × ℝ⁺, (1.1b) ℬ₁????(????, ????) − ∫₀⁺∞ ????′(????) ∂????[????(????, ????) − ????(????, ???? − ????)]d???? = 0, on Γ₀ × ℝ⁺, (1.1c) ℬ₂????(????, ????) + ∫₀⁺∞ ????′(????)[????(????, ????) − ????(????, ???? − ????)]d???? = ????(????, ????), on Γ₁ × ℝ⁺, (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>