General decay of solutions of a wave equation with memory term and acoustic boundary condition

Vicente, André; Frota, C.L.

doi:10.1002/mma.4129

“…The simplest case in which σ is constant and the operator div (σ ∇ ) reduces (up to σ ) to the Laplace-Beltrami operator was briefly considered in [7, §6] and then studied in [10,21,22,[49][50][51]. In these papers, the authors assume 0 = ∅ and the homogeneous boundary condition on it is replaced by the homogeneous Dirichlet boundary condition.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The damped wave equation with acoustic boundary conditions and non-locally reacting surfaces

Barbieri

¹

,

Vitillaro

²

2022

Semigroup Forum

1

0

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The aim of the paper is to study the problem $$\begin{aligned} {\left\{ \begin{array}{ll} u_{tt}+du_t-c^2\Delta u=0 \qquad &{}\text {in}\, {\mathbb {R}}\times \Omega ,\\ \mu v_{tt}- \textrm{div}_\Gamma (\sigma \nabla _\Gamma v)+\delta v_t+\kappa v+\rho u_t =0\qquad &{}\text {on}\, {\mathbb {R}}\times \Gamma _1,\\ v_t =\partial _\nu u\qquad &{}\text {on}\, {\mathbb {R}}\times \Gamma _1,\\ \partial _\nu u=0 &{}\text {on}\, {\mathbb {R}}\times \Gamma _0,\\ u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x) &{} \text {in}\, \Omega ,\\ v(0,x)=v_0(x),\quad v_t(0,x)=v_1(x) &{} \text {on}\, \Gamma _1, \end{array}\right. } \end{aligned}$$ u tt + d u t - c 2 Δ u = 0 in R × Ω , μ v tt - div Γ ( σ ∇ Γ v ) + δ v t + κ v + ρ u t = 0 on R × Γ 1 , v t = ∂ ν u on R × Γ 1 , ∂ ν u = 0 on R × Γ 0 , u ( 0 , x ) = u 0 ( x ) , u t ( 0 , x ) = u 1 ( x ) in Ω , v ( 0 , x ) = v 0 ( x ) , v t ( 0 , x ) = v 1 ( x ) on Γ 1 , where $$\Omega $$ Ω is a open domain of $${\mathbb {R}}^N$$ R N with uniformly $$C^r$$ C r boundary ($$N\ge 2$$ N ≥ 2 , $$r\ge 1$$ r ≥ 1 ), $$\Gamma =\partial \Omega $$ Γ = ∂ Ω , $$(\Gamma _0,\Gamma _1)$$ ( Γ 0 , Γ 1 ) is a relatively open partition of $$\Gamma $$ Γ with $$\Gamma _0$$ Γ 0 (but not $$\Gamma _1$$ Γ 1 ) possibly empty. Here $$\textrm{div}_\Gamma $$ div Γ and $$\nabla _\Gamma $$ ∇ Γ denote the Riemannian divergence and gradient operators on $$\Gamma $$ Γ , $$\nu $$ ν is the outward normal to $$\Omega $$ Ω , the coefficients $$\mu ,\sigma ,\delta , \kappa , \rho $$ μ , σ , δ , κ , ρ are suitably regular functions on $$\Gamma _1$$ Γ 1 with $$\rho ,\sigma $$ ρ , σ and $$\mu $$ μ uniformly positive, d is a suitably regular function in $$\Omega $$ Ω and c is a positive constant. In this paper we first study well-posedness in the natural energy space and give regularity results. Hence we study asymptotic stability for solutions when $$\Omega $$ Ω is bounded, $$\Gamma _1$$ Γ 1 is connected, $$r=2$$ r = 2 , $$\rho $$ ρ is constant and $$\kappa ,\delta ,d\ge 0$$ κ , δ , d ≥ 0 .

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“…Recently, wave equations with acoustic boundary conditions have been studied by many authors. [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] Compared with previous articles on this subject, the main difference of this article is related with Equations (1) 3 and (2) 3 . In most previous work, for instance, 9,[12][13][14]16,17,19 the memory term of 𝜕 𝜈 u t has not been considered.…”

Section: Introductionmentioning

confidence: 96%

“…[5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] Compared with previous articles on this subject, the main difference of this article is related with Equations (1) 3 and (2) 3 . In most previous work, for instance, 9,[12][13][14]16,17,19 the memory term of 𝜕 𝜈 u t has not been considered. In Liu and Chen, 13 they discussed the equation including the memory term for 𝜕 𝜈 u t on Γ 0 , and also with the viscoelastic term Δu t in the domain Ω. Liu and Sun 14 studied the following problem:…”

Section: Introductionmentioning

confidence: 96%

“…The asymptotic behavior was obtained in Beale 3 (Theorem 2.6). Recently, wave equations with acoustic boundary conditions have been studied by many authors 5–19 …”

Section: Introductionmentioning

confidence: 99%

“…Compared with previous articles on this subject, the main difference of this article is related with Equations () 3 and () 3 . In most previous work, for instance, 9,12–14,16,17,19 the memory term of ∂ ν u t has not been considered. In Liu and Chen, 13 they discussed the equation including the memory term for ∂ ν u t on Γ 0 , and also with the viscoelastic term Δ u t in the domain Ω. Liu and Sun 14 studied the following problem:

\{\begin{array}{l} u_{t t} - normalΔ u + α false(t false) {true\int}_{0}^{t} g false(t - s false) normalΔ u false(s false) d s = 0, in normalΩ \times false(0, \infty false), \\ u = 0, on {normalΓ}_{0} \times false(0, \infty false), \\ \partial_{ν} u - α false(t false) {true\int}_{0}^{t} g false(t - s false) \partial_{ν} u false(s false) d s = y_{t}, on {normalΓ}_{1} \times false(0, \infty false), \\ u_{t} + p false(x false) y_{t} + q false(x false) y = 0, on {normalΓ}_{1} \times false(0, \infty false), \\ u false(x, 0 false) = u_{0} false(x false), u_{t} false(x, 0 false) = u_{1} false(x false), y false(x, 0 false) = y_{0} false(x false), in normalΩ . \end{array}

With the perturbed energy functional technique, they established a general decay result relying on the behavior of...…”

Section: Introductionmentioning

confidence: 99%

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In this paper, we study the exponential decay of the energy associated to an initial value problem involving the wave equation on the hyperbolic space 𝔹 𝑁 . The space 𝔹 𝑁 is the unit disc {𝑥 ∈ ℝ 𝑁 ∶ |𝑥| < 1} of ℝ 𝑁 endowed with the Riemannian metric 𝑔 given by 𝑔 𝑖𝑗 = 𝑝 2 𝛿 𝑖𝑗 , where 𝑝(𝑥) = 2 1−|𝑥| 2 and 𝛿 𝑖𝑗 = 1, if 𝑖 = 𝑗 and 𝛿 𝑖𝑗 = 0, if 𝑖 ≠ 𝑗. Making an appropriate change, the problem can be seen as a singular problem on the boundary of the open ball 𝐵 1 = {𝑥 ∈ ℝ 𝑁 ; |𝑥| < 1} endowed with the euclidean metric. The proof is based on the multiplier techniques combined with the use of Hardy's inequality, in a version due to the Brezis-Marcus, which allows us to overcome the difficulty involving the singularities.

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General decay of solutions of a wave equation with memory term and acoustic boundary condition

Cited by 14 publications

References 30 publications

The damped wave equation with acoustic boundary conditions and non-locally reacting surfaces

The damped wave equation with acoustic boundary conditions and non-locally reacting surfaces

Exponential stabilization of wave equation with acoustic boundary conditions and its application to memory‐type boundary

Exponential decay for semilinear wave equation with localized damping in the hyperbolic space

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