Stability of planar stationary waves for damped wave equations with nonlinear convection in multi-dimensional half space
Stability of planar stationary waves for damped wave equations with nonlinear convection in multi-dimensional half space
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DOI：
10.3934/krm.2008.1.49
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发表时间：
2008-02
影响因子：
1
通讯作者：
Yoshihiro Ueda;Tohru Nakamura;S. Kawashima
中科院分区：
文献类型：
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作者：
Yoshihiro Ueda;Tohru Nakamura;S. Kawashima
本文研究了多维半空间中带非线性对流项的阻尼波动方程初边值问题解的大时间性态。我们证明了在初始扰动很小的条件下，当时间趋于无穷大时，问题的解收敛到相应的平面驻波。证明了解的切向导数验证了$t\to\infty$的定量衰减估计。此外，通过假设初始扰动在法线方向上代数衰减，得到了一个额外的代数收敛速度。证明的关键是利用时空加权能量方法得到解的先验估计。
In this paper, we consider the large-time behavior of solutions to the initial-boundary value problem for damped wave equations with a nonlinear convection term in the multi-dimensional half space. We show that the solution to the problem converges to the corresponding planar stationary wave as time tends to infinity under smallness condition on the initial perturbation. It is proved that the tangential derivatives of the solution verify quantitative decay estimates for $t\to\infty$. Moreover, an additional algebraic convergence rate is obtained by assuming that the initial perturbation decays algebraically in the normal direction. The crucial point of the proof is to derive a priori estimates of solutions by using the time and space weighted energy method.