带有非有效且非散射产生阻尼的半线性波动方程的Strauss指数与Glassey指数
Strauss and Glassey exponents for semilinear wave equations with a non-effective and not scattering producing damping
AI总结：
本文针对带非有效阻尼的半线性波动方程，在亚Strauss和亚Glassey范围分别证明|u|^p和|u_t|^p型非线性弱解的爆破，推导了对应迭代框架并利用比较论证完成证明。
AI中文摘要：
本文研究带有随时间变化系数的半线性阻尼波动方程的柯西问题，该阻尼项属于具有临界衰减率（含迭代对数因子）的非有效阻尼项类别。非线性项考虑了|u|^p和|u_t|^p两种形式。假设初始数据非负且具有紧支集，对于非线性项|u|^p，我们在亚Strauss范围1 < p ≤ p_{Str}(n)内建立弱解的爆破结果。亚临界情形的证明依赖于解的空间平均的迭代框架，该框架通过利用与阻尼项系数相关的随时间变化的乘子得到。另一方面，在极限情形p = p_{Str}(n)下，我们处理具有分离变量的齐次方程的解，并研究相应随时间变化常微分方程的基本解系的性质，以推导解的合适加权空间平均的迭代框架。最后，对于导数型非线性项|u_t|^p，我们通过使用与对应局部时间解相关的合适随时间变化函数的比较论证，在亚Glassey范围1 < p ≤ (n+1)/(n-1)内证明弱解的爆破结果。
英文摘要：
In this paper, we study the Cauchy problems for a semilinear damped wave equation with a time-dependent coefficient for the damping term belonging to the class of non-effective damping terms with critical decay rates involving iterated logarithmic factors. As nonlinearities we consider both $|u|^p$ and $|u_t|^p$. Assuming nonnegative and compactly supported initial data, we establish the blow-up for weak solutions in the sub-Strauss range $1 < p \leq p_{\mathrm{Str}}(n)$ for the power of the nonlinear term $|u|^p$. The proof in the sub-critical case relies on an iteration frame for the space average of the solution, obtained by employing a time-dependent multiplier related to the coefficient of the damping term. On the other hand, in the limit case $p=p_{\mathrm{Str}}(n)$, we work with solutions of the homogeneous equation with separated variables, and we investigate the properties of a fundamental system of solutions for the corresponding time-dependent ODE, in order to derive an iteration frame for a suitable weighted space average of the solution. Finally, for the derivative type nonlinearity $|u_t|^p$ we prove the blow-up of weak solutions in the sub-Glassey range $1 < p \leq \frac{n+1}{n-1}$ by using a comparison argument for a suitable time-dependent function associated with the corresponding local in time solution.
