AI 中文总结
本文证明二维外域中带导数型幂次非线性项的半线性阻尼波动方程的小初值解时间整体存在，通过构造时间加权函数空间与非线性估计完成证明，并讨论解的时间导数的长时间行为。
AI 中文摘要
本文研究二维外域中带导数型幂次非线性项的半线性阻尼波动方程，证明其小初值解的时间整体存在性，该结果此前未见相关研究。证明基于构造合适的时间加权函数空间，并推导与Dirichlet拉普拉斯分数次幂兼容的非线性估计，以匹配半群衰减结构。此外，本文还讨论所得整体解的时间导数的长时间行为及其精确估计。
英文摘要
In this paper, we are interested in considering the semi-linear damped wave equation with the power nonlinearity of derivative type in $2$D exterior domains to indicate the global (in time) existence of small data solutions, which has never appeared in previous studies. Our proof is based on constructing a judicious time-weighted function space and demonstrating nonlinear estimates compatible with fractional powers of the Dirichlet Laplacian linked to the semigroup decay structure. Moreover, the large time behavior of the time derivative of the obtained global solutions and its sharp estimate are also discussed in this paper.
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