论文标题
关于结构声学模型的解决方案的渐近行为
On the asymptotic behavior of solutions to a structure acoustics model
论文作者
论文摘要
本文涉及解决结构声模型的长期行为，该结构声学模型由在平稳的有限域$ω\ subset \ mathbb {r}^3 $上定义的半连续波方程组成，该方程与$ω$ $ω$的边界的平坦部分作用于berger板方程。该系统受几个竞争力的影响，特别是在波动方程上作用的源术语，该术语允许具有超临界指数。 我们的结果基于贝克林和拉马哈[8]获得的结果。有了对系统中参数的一些限制，并且在涉及NEHARI歧管的仔细分析中，我们获得了潜在井溶液的全球存在，并建立了指数或代数衰减的能量速率，取决于阻尼项的行为。这项工作的主要新颖性在于我们的稳定估计，这尤其不是产生低阶项。因此，主要结果的证明是更短，更简洁。
This article concerns the long term behavior of solutions to a structural acoustic model consisting of a semilinear wave equation defined on a smooth bounded domain $Ω\subset\mathbb{R}^3$ which is coupled with a Berger plate equation acting on a flat portion of the boundary of $Ω$. The system is influenced by several competing forces, in particular a source term acting on the wave equation which is allowed to have a supercritical exponent. Our results build upon those obtained by Becklin and Rammaha [8]. With some restrictions on the parameters in the system and with careful analysis involving the Nehari manifold we obtain global existence of potential well solutions and establish either exponential or algebraic decay rates of energy, dependent upon the behavior of the damping terms. The main novelty in this work lies in our stabilization estimate, which notably does not generate lower-order terms. Consequently, the proof of the main result is shorter and more concise.
