发表机构
University of Boumerdes; Universität Paderborn（布迈尔迪斯大学; 帕德博恩大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对Kuznetsov与Westervelt型粘性波动方程，证明在一阶Sobolev空间小初值条件下，满足特定非线性项条件时存在全局经典解且解指数衰减，填补了相关研究空白。
AI 中文摘要
在光滑有界区域$\boldsymbol{\textit{\u03a9}} \boldsymbol{\boldsymbol{\u2282}} \boldsymbol{\textit{\u00a1}}^n$（其中$n \boldsymbol{\u2265} 1$，$a>0$）中，我们研究一般粘性波动方程的初边值问题，该方程为$\boldsymbol{\textit{\u0068}}(\boldsymbol{\textit{\u0075}},\boldsymbol{\textit{\u0075}}_{\boldsymbol{\textit{\u0074}}}) \boldsymbol{\textit{\u0075}}_{\boldsymbol{\textit{\u0074\u0074}}} \boldsymbol{\boldsymbol{\u003d}} \boldsymbol{\u0394} \boldsymbol{\textit{\u0075}}_{\boldsymbol{\textit{\u0074}}} \boldsymbol{\u002b} \boldsymbol{\textit{\u0061}} \boldsymbol{\u0394} \boldsymbol{\textit{\u0075}} \boldsymbol{\u002b} \boldsymbol{\textit{\u0066}}(\boldsymbol{\textit{\u0075}},\boldsymbol{\textit{\u0075}}_{\boldsymbol{\textit{\u0074}}},\boldsymbol{\u2207} \boldsymbol{\textit{\u0075}},\boldsymbol{\u2207} \boldsymbol{\textit{\u0075}}_{\boldsymbol{\textit{\u0074}}})$，该方程出现在非线性声学波传播模型中，已确立的Kuznetsov型和Westervelt型方程是其特殊例子。现有文献对二阶及更高阶Sobolev空间中足够小初值$(\boldsymbol{\textit{\u0075}}_{\boldsymbol{\textit{\u0030}}}, \boldsymbol{\textit{\u0075}}_{\boldsymbol{\textit{\u0030\u0074}}}) \boldsymbol{\boldsymbol{\u003d}} (\boldsymbol{\textit{\u0075}}, \boldsymbol{\textit{\u0075}}_{\boldsymbol{\textit{\u0074}}})|_{\boldsymbol{\textit{\u0074}} \boldsymbol{\boldsymbol{\u003d}} \boldsymbol{\textit{\u0030}}}$的全局解提供了大量结果，但仅涉及一阶Sobolev空间的小性条件下全局可解性能否成立仍悬而未决。本手稿针对该问题，证明当非线性项$\boldsymbol{\textit{\u0068}}$和$\boldsymbol{\textit{\u0066}}$足够光滑，且满足$\boldsymbol{\textit{\u0068}}(0,0)\boldsymbol{\boldsymbol{\u003e}}0$、$\boldsymbol{\textit{\u0066}}(0,0,0,0)\boldsymbol{\boldsymbol{\u003d}}0$及$\boldsymbol{\u2207} \boldsymbol{\textit{\u0066}}(0,0,0,0)\boldsymbol{\boldsymbol{\u003d}}0$时，$(\boldsymbol{\textit{\u0075}},\boldsymbol{\textit{\u0075}}_{\boldsymbol{\textit{\u0074}}})$在$W^{1,r}\times W^{1,p}$-Sobolev空间中存在全局经典解且指数衰减。
英文摘要
In a smoothly bounded domain $\Om\subset\R^n$ with $n\geq 1$ and $a>0$, we consider an initial-boundary value problem for the general viscous wave equation \bas h(u,u_t) u_{tt} = \Del u_t + a\Del u + f(u,u_t,\na u,\na u_t) \eas which appears in models of nonlinear acoustics wave propagation; well-established equations of Kuznetsov and Westervelt type form particular examples.\abs % While the existing literature offers extensive results on global solutions for sufficiently small initial data $(u_{0}, u_{0t})=(u, u_{t})|_{t=0}$ in second- and higher-order Sobolev spaces it appears to remain open how far global solvability can be established under smallness conditions involving only first-order Sobolev spaces. The present manuscript addresses this question by proving the existence of global classical solutions together with exponential decay of the pair $(u,u_{t})$ in $ W^{1,r}\times W^{1,p}$-Sobolev spaces whenever the nonlinearities $h$ and $f$ are sufficiently smooth and are such that $h(0,0)>0$ as well as $f(0,0,0,0)=0$ and $\na f(0,0,0,0)=0$.
