DECAY PROPERTY OF REGULARITY-LOSS TYPE AND APPLICATION TO SOME NONLINEAR HYPERBOLIC-ELLIPTIC SYSTEM
DECAY PROPERTY OF REGULARITY-LOSS TYPE AND APPLICATION TO SOME NONLINEAR HYPERBOLIC-ELLIPTIC SYSTEM
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DOI：
10.1142/s021820250600173x
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发表时间：
2006
影响因子：
3.5
通讯作者：
Takafumi Hosono;S. Kawashima
中科院分区：
文献类型：
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作者：
Takafumi Hosono;S. Kawashima
讨论了一类具有四阶椭圆部分的非线性双曲椭圆方程组Cauchy问题的整体可解性和解的渐近性态。该系统是最简单的辐射气体模型的一个改进版本，并验证了规则性损失型的衰减特性。这样的耗散结构也出现在里维拉和Racke研究的耗散的Krishenko系统中。这种耗散性在高频区域非常弱，导致难以得到非线性问题全局解的先验估计。事实证明，通常的能量方法并不奏效。我们克服了这个困难，采用时间加权能量方法，结合最佳衰减的低阶导数的解决方案，我们建立了一个整体存在性和渐近衰减的结果。此外，我们表明，该解决方案有一个渐进的自相似的配置文件所描述的Burgers方程的时间趋于无穷大。
We discuss the global solvability and asymptotic behavior of solutions to the Cauchy problem for some nonlinear hyperbolic–elliptic system with a fourth-order elliptic part. This system is a modified version of the simplest radiating gas model and verifies a decay property of regularity-loss type. Such a dissipative structure also appears in the dissipative Timoshenko system studied by Rivera and Racke. This dissipative property is very weak in high frequency region and causes the difficulty in deriving the desired a priori estimates for global solutions to the nonlinear problem. In fact, it turns out that the usual energy method does not work well. We overcome this difficulty by employing a time-weighted energy method which is combined with the optimal decay for lower order derivatives of solutions, and we establish a global existence and asymptotic decay result. Furthermore, we show that the solution has an asymptotic self-similar profile described by the Burgers equation as time tends to infinity.