论文标题
具有声学边界条件和非局部反应表面的阻尼波方程
The Damped Wave Equation with Acoustic Boundary Conditions and Non-locally Reacting Surfaces
论文作者
论文摘要
本文的目的是研究问题$$ u_ {tt}+du_t-c^2ΔU= 0 \ qquad \ text {in $ \ m arthbb {r} \timsomΩ$，} $ $ $ $ $ $ $μv_{tt} - = 0 \ qquad \ text {on $ \ mathbb {r} \ timesγ_1$，} $$ $$ $$ v_t = \partial_νu\ qquad \ qquad \ qquad \ text {on $ \ m \ mathbb {r} \ timesγ_1γ_1γ_1$，} γ_0$，} $$ $$ u（0，x）= u_0（x），\ quad u_t（0，x）= u_1（x）\ quad \ text {in $ω$，} $ω$，} $ $ $ $ $ v（0，x）= v_0（x），\ quad v_t（x） $γ_1$，} $$其中$ω$是$ \ mathbb {r}^n $的开放域，均匀地$ c^r $界限（$ n \ ge 2 $，$ r \ ge 1 $），$γ= \partialΩ可能是空的。这里$ \ text {div}_γ$和$ \nabla_γ$表示Riemannian的分歧和梯度运营商在$γ$，$γ$，$ν$上是正常向外的$ω$，系数$μ，σ，σ，δ，δ，κ，ρ$适合$ $ $ $ $ $ quy $和$γ_1$和$ c，$ unive，$ unive $ un $ω$和$ c $的适当常规功能是一个正常数。 在本文中，我们首先在自然能量空间中研究了良好的性能，并给出规律性的结果。因此，我们研究解决方案的渐近稳定性，当$ω$有限，$γ_1$连接，$ r = 2 $，$ρ$是常数和$κ，δ，d \ ge 0 $。
The aim of the paper is to study the problem $$u_{tt}+du_t-c^2Δu=0 \qquad \text{in $\mathbb{R}\timesΩ$,}$$ $$μv_{tt}- \text{div}_Γ(σ\nabla_Γv)+δv_t+κv+ρu_t =0\qquad \text{on $\mathbb{R}\times Γ_1$,}$$ $$v_t =\partial_νu\qquad \text{on $\mathbb{R}\times Γ_1$,}$$ $$\partial_νu=0 \text{on $\mathbb{R}\times Γ_0$,}$$ $$u(0,x)=u_0(x),\quad u_t(0,x)=u_1(x)\quad \text{in $Ω$,}$$ $$v(0,x)=v_0(x),\quad v_t(0,x)=v_1(x) \quad \text{on $Γ_1$,}$$ where $Ω$ is a open domain of $\mathbb{R}^N$ with uniformly $C^r$ boundary ($N\ge 2$, $r\ge 1$), $Γ=\partialΩ$, $(Γ_0,Γ_1)$ is a relatively open partition of $Γ$ with $Γ_0$ (but not $Γ_1$) possibly empty. Here $\text{div}_Γ$ and $\nabla_Γ$ denote the Riemannian divergence and gradient operators on $Γ$, $ν$ is the outward normal to $Ω$, the coefficients $μ,σ,δ, κ, ρ$ are suitably regular functions on $Γ_1$ with $ρ,σ$ and $μ$ uniformly positive, $d$ is a suitably regular function in $Ω$ 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 $Ω$ is bounded, $Γ_1$ is connected, $r=2$, $ρ$ is constant and $κ,δ,d\ge 0$.
