Smooth solutions to the heat equation which are nowhere analytic in time

TL;DR

This paper demonstrates that smooth solutions to the heat equation can be constructed to be nowhere analytic in time, especially in domains with general boundary conditions or without growth restrictions, challenging previous assumptions.

Contribution

It constructs explicit examples of bounded and exponentially growing solutions to the heat equation that are nowhere analytic in time, showing time analyticity does not always hold.

Findings

Constructed bounded solutions in the half plane that are nowhere analytic in time.

Found solutions with exponential growth of order 2+δ that are nowhere analytic in time.

Showed time analyticity fails in general domains with certain boundary conditions.

Abstract

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 $\delta>0$, we find a solution to the heat equation on the whole plane, with exponential growth of order $2+\delta$, which is nowhere analytic in time.