A General Decay Result for a Von Karman Equation With Memory and Acoustic Boundary Conditions

A general stability result for a viscoelastic von Karman equation

“…where R + = (0, ∞), Ω is a bounded domain in R 2 with a smooth boundary 𝜕Ω, 𝜈 is the unit normal vector outward to 𝜕Ω, [u, z] = u x 1 x 1 z x 2 x 2 + u x 2 x 2 z x 1 x 1 − 2u x 1 x 2 z x 1 x 2 for (x 1 , x 2 ) ∈ Ω, and 𝜇 is a kernel function with certain properties. Many researchers have expanded stability results for viscoelastic von Karman equations [1][2][3][4][5]. Munoz Rivera and Menzala [1] obtained an exponential decay result for problem (1.1)- (1.4) with the rotational inertia (−aΔu tt ) when 𝜇 ∶ [0, ∞) → (0, ∞) verifies −𝜁 1 𝜇(t) ≤ 𝜇 ′ (t) ≤ −𝜁 2 𝜇(t), 0 ≤ 𝜇 ′′ (t) ≤ 𝜁 3 𝜇(t), (1.5) where 𝜁 1 , 𝜁 2 , 𝜁 3 > 0.…”

“…Many researchers have expanded stability results for viscoelastic von Karman equations [1–5]. Munoz Rivera and Menzala [1] obtained an exponential decay result for problem ()–() with the rotational inertia ( $$$ -a\Delta {u}_{tt} $$$) when $$$ \mu :\left[0,\infty \right)\to \left(0,\infty \right) $$$ verifies $$$$ -{\zeta}_1\mu (t)\le {\mu}^{\prime }(t)\le -{\zeta}_2\mu (t),0\le {\mu}^{\prime \prime }(t)\le {\zeta}_3\mu (t), $$$$ where $$$ {\zeta}_1,{\zeta}_2,{\zeta}_3>0.$…”