论文标题
粘性耗散对具有第二个声音的非线性声波方程的影响
The influence of viscous dissipations on the nonlinear acoustic wave equation with second sound
论文作者
论文摘要
我们研究粘性耗散对非线性声学中Cattaneo型模型库奇问题的影响，该模型是通过将Lighthill近似值用于粘性或不粘性流体模型来确定的。本文的贡献是双重的。对于非线性粘性Cattaneo型模型，涉及粘性阻尼的分数laplacian $（-Δ）^α$，并在[0,1] $中$α\ in [0,1] $中，我们在某些Sobolev空间中具有小数据的全球（及时）解决方案的最佳衰减率。此外，通过引入阈值$α= 1/2 $，用于分数粘性耗散的功率，当$α\ in [0,1/2）$ in [0,1/2）$中时得出异常的扩散曲线，并且当[1/2,1] $ in [0,1/2）$ in [0,1/2）$中得出了$α\。尽管对于非线性无粘性Cattaneo型模型（或关键情况下的Jordan-Moore-Gibson-Thompson方程），我们在适当的假设下以适当的假设获得了能量解决方案的爆炸。因此，在非线性cattaneo型模型中存在粘性耗散的存在是溶液的全局（时代）存在和爆炸的标准。
We study the effect of a viscous dissipation on the Cauchy problem for a Cattaneo-type model in nonlinear acoustics, established by applying the Lighthill approximation for the viscous or inviscid fluid model. The contribution of this paper is twofold. For the nonlinear viscous Cattaneo-type model involving a fractional Laplacian $(-Δ)^α$ in the viscous damping with $α\in[0,1]$, we derive optimal decay rates for global (in time) solutions with small data in certain Sobolev spaces. Furthermore, by introducing a threshold $α=1/2$ for the power of the fractional viscous dissipation, we derive an anomalous diffusion profile when $α\in[0,1/2)$ and a diffusion wave profile when $α\in[1/2,1]$ for large-time. Whereas, for the nonlinear inviscid Cattaneo-type model (or the Jordan-Moore-Gibson-Thompson equation in the critical case), we obtain the blow-up of the energy solutions in finite time under suitable assumptions for the initial data. Thus, the presence of a viscous dissipation in the nonlinear Cattaneo-type model is a criterion for the global (in time) existence and blow-up of solutions.
