In this paper, we study the initial value problem for semilinear wave equations with the time-dependent and scale-invariant damping in two dimensions. Similarly to the one dimensional case by Kato, Takamura and Wakasa in 2019, we obtain the lifespan estimates of the solution for a special constant in the damping term, which are classified by total integral of the sum of the initial position and speed. The key fact is that, only in two space dimensions, such a special constant in the damping term is a threshold between "wave-like" domain and "heat-like" domain. As a result, we obtain a new type of estimate especially for the critical exponent.

中文翻译：

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二维空间尺度不变阻尼半线性波动方程解的寿命

在本文中，我们研究了具有二维瞬态和尺度不变阻尼的半线性波动方程的初值问题。与 Kato、Takamura 和 Wakasa 在 2019 年的一维案例类似，我们获得了阻尼项中一个特殊常数的解的寿命估计，它们通过初始位置和速度之和的总积分进行分类。关键的事实是，只有在二维空间中，阻尼项中的这种特殊常数才是“类波”域和“类热”域之间的阈值。结果，我们获得了一种新的估计类型，特别是对于临界指数。