heat equation的光滑解在任何地方都不是时间解析的
Smooth solutions to the heat equation which are nowhere analytic in time
摘要：
光滑但无处解析函数的存在是众所周知的（du Bois-Reymond，Math. Ann.，21（1）：109-117，1883）。然而，热方程的光滑解通常是空间变量的解析解。众所周知（Kowalevsky，Crelle，80：1-32，1875），热方程的解在$t=0$上可能不是时间解析的，即使初始函数是实解析的。最近，它表明在\cite{Zha20, DZ20, DP20}，解决方案的热方程在整个空间，或半空间与零边界值，是解析的时间在本质上最优的条件。在本文中，我们表明，时间解析并不总是真的，在一般的边界条件或没有合适的增长条件的域。更精确地说，我们构造了热方程在半平面中的两个有界解，它们在时间上无处解析。此外，对于任何$δ>0$，我们找到一个解决方案的热方程在整个平面上，指数增长性的顺序$2+δ$，这是无处解析的时间。
英文摘要：
The existence of smooth but nowhere analytic functions is well-known (du Bois-Reymond, Math. Ann., 21(1):109-117, 1883). However, smooth solutions to the heat equation are usually analytic in the space variable. It is also well-known (Kowalevsky, Crelle, 80:1-32, 1875) that a solution to the heat equation may not be time-analytic at $t=0$ even if the initial function is real analytic. Recently, it was shown in \cite{Zha20, DZ20, DP20} that solutions to the heat equation in the whole space, or half space with zero boundary value, are analytic in time under essentially optimal conditions. In this paper, we show that time analyticity is not always true in domains with general boundary conditions or without suitable growth conditions. More precisely, we construct two bounded solutions to the heat equation in the half plane which are nowhere analytic in time. In addition, for any $δ>0$, we find a solution to the heat equation on the whole plane, with exponential growth of order $2+δ$, which is nowhere analytic in time.
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