具时变阻尼和质量项的半线性波动方程的爆破准则与寿命估计
Blow-up criteria and lifespan estimates for semilinear wave equations with time-dependent damping and mass
AI总结：
该研究针对带时变阻尼、质量项及导数型非线性项的半线性波动方程，通过Liouville变换等方法推导了能量解的爆破准则与寿命估计，覆盖质量主导及阻尼扰动抵消的情形。
AI中文摘要：
我们研究带有时间依赖阻尼、质量项以及导数型非线性项的半线性波动方程。通过应用Liouville变换并求解从无穷远处建立的Volterra积分方程，我们构造了伴随方程的正精确解，而无需使用线性传播子的显式表示。该解用于推导具有有限传播性的能量解的爆破准则，以及寿命的上界。若阻尼系数的原函数增长速度至多为对数级，我们得到了完整的移位Glassey范围，包括临界指数，以及相应的多项式和指数寿命估计。该结果与尺度不变判别式的符号无关，因此包含质量主导的情况。当阻尼系数的非可积振荡扰动的贡献被对应的质量项抵消时，该结果也适用。
英文摘要：
We consider semilinear wave equations with time-dependent damping and mass and a derivative-type nonlinearity. By applying a Liouville transformation and solving a Volterra integral equation posed from infinity, we construct a positive exact solution to the adjoint equation without using an explicit representation of the linear propagator. This solution is used to derive a blow-up criterion for energy solutions with finite propagation, together with an upper bound for the lifespan. If the primitive of the damping coefficient grows at most logarithmically, we obtain the full shifted Glassey range, including the critical exponent, and the corresponding polynomial and exponential lifespan estimates. The result applies independently of the sign of the scale-invariant discriminant and therefore includes the mass-dominant regime. It also covers non-integrable oscillatory perturbations of the damping coefficient when their contributions are canceled by the corresponding mass terms.
