AI 中文总结
研究二维和三维空间中的中尾问题，通过结合阻尼与无阻尼分量的相关公式及论证方法，建立小数据温和Sobolev解的全局时间存在性与唯一性，还得到相关估计，三维中避免对\(p\)的限制，且无阻尼分量散射到自由波。
AI 中文摘要
我们研究二维和三维空间中的中尾问题，它是一个由半线性阻尼波动方程和半线性无阻尼波动方程组成的弱耦合系统。我们在新的允许范围内建立了小数据温和Sobolev解的全局时间存在性和唯一性，以及与相应线性化问题相匹配的时间相关估计。证明将阻尼分量的扩散型\(L^m - L^r\)估计与无阻尼分量的泊松公式和基尔霍夫公式相结合。引入了依赖于维度的解空间以纳入二维空间中较弱的波衰减和对数\(L^2\)增长。在三维空间中，两级不动点论证通过在强空间中建立自映射性质和在较弱度量下的收缩来避免对\(p\)的人为限制。此外，无阻尼分量在\(H^2\times H^1\)中散射到自由波。
英文摘要
We study Nakao's problem in two and three space dimensions, a weakly coupled system consisting of a semilinear damped wave equation and a semilinear undamped wave equation. We establish the global in-time existence and uniqueness of small data mild Sobolev solutions in new admissible ranges, together with time-dependent estimates matching those for the corresponding linearized problems. The proof combines diffusion-type $L^m-L^r$ estimates for the damped component with the Poisson and Kirchhoff formulas for the undamped component. Dimension-dependent solution spaces are introduced to incorporate the weaker wave decay and logarithmic $L^2$ growth in two space dimensions. In three space dimensions, a two-level fixed point argument avoids an artificial restriction on $p$ by establishing the self-map property in a strong space and the contraction in a weaker metric. Furthermore, the undamped component scatters to a free wave in $H^2\times H^1$.
