Abstract
In this paper, we are concerned with the energy decay rate of the nonlinear viscoelastic problem with dynamic and acoustic boundary conditions.
Highlights
In this paper, we are concerned with the energy decay rate of the following nonlinear viscoelastic problem with a time-varying delay in the boundary feedback and acoustic boundary conditions:t utt – δ0 u + g(t – s) div a(x)∇u(s) ds + δ1 + b(x) ut(t) m–2 ut= |u|p–2u in × (0, +∞), (1.1)u = 0 on 0 × (0, ∞), (1.2)∂ u(t) utt + δ0 ∂ν – t g(t – s) a(x)∇u(s)· ν ds + μ1k1 ut(t)+ μ2k2 ut t – τ (t)= h(x)yt on 1 × (0, ∞), (1.3)
Liu [29] investigated the following viscoelastic wave equation with an interval timevarying delay term: t utt(x, t) – u(x, t) + α(t) g(t – s) u(x, s) ds + a0ut(x, t) + a1 x, t – τ (t) = 0 in × (0, ∞), u(x, t) = 0 on ∂ × (0, ∞), u(x, 0) = u0(x), ut(x, 0) = u1(x) in, ut x, t – τ (0) = f0 x, t – τ (0) on × 0, τ (0), where is a bounded domain Rn (n ≥ 2) with a boundary ∂ of class C2, α and g are positive non-increasing functions defined on R+, a0 and a1 are real number with a0 > 0, τ (t) > 0 represents the time-varying delay
In this paper, we study the energy decay rate of the nonlinear viscoelastic problem with a time-varying delay in the boundary conditions
Summary
We are concerned with the energy decay rate of the following nonlinear viscoelastic problem with a time-varying delay in the boundary feedback and acoustic boundary conditions:. T utt(x, t) – u(x, t) + α(t) g(t – s) u(x, s) ds + a0ut(x, t) + a1 x, t – τ (t) = 0 in × (0, ∞), u(x, t) = 0 on ∂ × (0, ∞), u(x, 0) = u0(x), ut(x, 0) = u1(x) in , ut x, t – τ (0) = f0 x, t – τ (0) on × 0, τ (0) , where is a bounded domain Rn (n ≥ 2) with a boundary ∂ of class C2, α and g are positive non-increasing functions defined on R+, a0 and a1 are real number with a0 > 0, τ (t) > 0 represents the time-varying delay He proved the general decay rate for the energy of a weak viscoelastic wave equation with an interval time-varying delay term.
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