非柱形时空集合中的线性非散度型抛物方程
Linear non-divergence parabolic equations in non-cylindrical space-time sets
AI总结：
该研究针对非柱形时空集合中的线性非散度型抛物方程，推导齐次问题解的多类衰减率、非齐次问题解的全时间与渐近估计，采用适配直径增长的非恒定时间步长迭代方法完成分析。
AI中文摘要：
我们在关于空间和时间变量均为非柱形的一般集合中，研究二阶线性非散度型抛物方程。该集合在任意有界时间区间上的限制是有界的，但当时间趋于无穷时，每个固定时间截面的空间直径可无界。对于齐次问题，我们在不同情形下得到了解的指数衰减、幂衰减及其他中间衰减率；对于非齐次问题，我们根据抛物边界上的数据和强迫函数，得到了解的全时间估计及时间趋于无穷时的渐近估计。该分析需要一般时空集合的微妙性质，以及用于提升增长引理的迭代格式，此类迭代的时间步长无需恒定，且适配于直径的增长。
英文摘要：
We study linear parabolic equations of the second order in non-divergence form in a general set which is non-cylindrical with respect to the spatial and time variables. The restriction of the set on any bounded time interval is bounded, but the spatial diameter of each fixed-time cross section can be unbounded as time tends to infinity. For homogeneous problems, we obtain exponential, power and other intermediate decaying rates for the solutions in different scenarios. For inhomogeneous problems, we obtain all-time and asymptotic, as time tend to infinity, estimates for the solutions in terms of the data on the parabolic boundary and forcing functions. The analysis requires subtle properties of general space-time sets and an iteration scheme to bootstrap the Growth Lemma. The time steps for such an iteration need not be constant and are adapted to the growth of the diameter.
