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Longtime Behavior of Wave Equation with Kinetic Boundary Condition
Applied Mathematics and Optimization ( IF 1.8 ) Pub Date : 2020-11-08 , DOI: 10.1007/s00245-020-09730-y
Xiaoyu Fu , Lingxia Kong

In this paper, we consider a damped wave equation with mixed boundary conditions in a bounded domain. On one portion of the boundary, we have kinetic boundary condition: \(\partial _\nu y+m(x)y_{tt}-\Delta _{T} y=0\) with density function m(x), and on the other portion, we have homogeneous Neumann boundary condition: \(\partial _\nu y=0\). Based on the growth of the resolvent operator on the imaginary axis, solutions of the wave equations under consideration are proved to decay logarithmically. The proof of the resolvent estimation relies on the interpolation inequalities for an elliptic equation with Steklov type boundary conditions.



中文翻译:

具有动力学边界条件的波动方程的长期行为

在本文中,我们考虑在有界域中具有混合边界条件的阻尼波方程。在边界的一部分上,我们具有动力学边界条件:\(\ partial _ \ nu y + m(x)y_ {tt}-\ Delta _ {T} y = 0 \),其密度函数为mx),在另一部分,我们有齐次的Neumann边界条件:\(\ partial _ \ nu y = 0 \)。基于分解算子在虚轴上的增长,证明所考虑的波动方程的解对数衰减。可分辨估计的证明依赖于具有Steklov型边界条件的椭圆方程的插值不等式。

更新日期:2020-11-09
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