In this paper, we investigate the pointwise time analyticity of three differential equations. They are the biharmonic heat equation, the heat equation with potentials and some nonlinear heat equations with power nonlinearity of order p. The potentials include all the nonnegative ones. For the first two equations, we prove if u satisfies some growth conditions in (x, t) ∈ M × [0, 1], then u is analytic in time (0, 1]. Here M is R d or a complete noncompact manifold with Ricci curvature bounded from below by a constant. Then we obtain a necessary and sufficient condition such that u(x, t) is analytic in time at t = 0. Applying this method, we also obtain a necessary and sufficient condition for the solvability of the backward equations, which is ill-posed in general.For the nonlinear heat equation with power nonlinearity of order p, we prove that a solution is analytic in time t ∈ (0, 1] if it is bounded in M × [0, 1] and p is a positive integer. In addition, we investigate the case when p is a rational number with a stronger assumption 0 < C 3 ≤ |u(x, t)| ≤ C 4 . It is also shown that a solution may not be analytic in time if it is allowed to be 0. As a lemma, we obtain an estimate of ∂ k t Γ(x, t; y) where Γ(x, t; y) is the heat kernel on a manifold, with an explicit estimation of the coefficients.An interesting point is that a solution may be analytic in time even if it is not smooth in the space variable x, implying that the analyticity of space and time can be independent. Besides, for general manifolds, space analyticity may not hold since it requires certain bounds on curvature and its derivatives.

In this paper, we investigate pointwise time analyticity of solutions to fractional heat equations in the settings of R d and a complete Riemannian manifold M. On one hand, in R d , we prove that any solution u = u (t, x) We also obtain pointwise estimates for ∂ k t p α (t, x; y), where p α (t, x; y) is the fractional heat kernel. Furthermore, under the same growth condition, we show that the mild solution is the unique solution. On the other hand, in a manifold M, we also prove the time analyticity of the mild solution under the same growth condition and the time analyticity of the fractional heat kernel, when M satisfies the Poincaré inequality and the volume doubling condition. Moreover, we also study the time and space derivatives of the fractional heat kernel in R d using the method of Fourier transform and contour integrals. We find that when α ∈ (0, 1], the fractional heat kernel is time analytic at t = 0 when x 0, which differs from the standard heat kernel.As corollaries, we obtain sharp solvability condition for the backward fractional heat equation and time analyticity of some nonlinear fractional heat equations with power nonlinearity of order p. These results are related to those in [8] and [21] which deal with local equations.

<p style='text-indent:20px;'>In this paper, we investigate the pointwise time analyticity of three differential equations. They are the biharmonic heat equation, the heat equation with potentials and some nonlinear heat equations with power nonlinearity of order <inline-formula><tex-math id="M1">\begin{document}$ p $\end{document}</tex-math></inline-formula>. The potentials include all the nonnegative ones. For the first two equations, we prove if <inline-formula><tex-math id="M2">\begin{document}$ u $\end{document}</tex-math></inline-formula> satisfies some growth conditions in <inline-formula><tex-math id="M3">\begin{document}$ (x,t)\in \mathrm{M}\times [0,1] $\end{document}</tex-math></inline-formula>, then <inline-formula><tex-math id="M4">\begin{document}$ u $\end{document}</tex-math></inline-formula> is analytic in time <inline-formula><tex-math id="M5">\begin{document}$ (0,1] $\end{document}</tex-math></inline-formula>. Here <inline-formula><tex-math id="M6">\begin{document}$ \mathrm{M} $\end{document}</tex-math></inline-formula> is <inline-formula><tex-math id="M7">\begin{document}$ R^d $\end{document}</tex-math></inline-formula> or a complete noncompact manifold with Ricci curvature bounded from below by a constant. Then we obtain a necessary and sufficient condition such that <inline-formula><tex-math id="M8">\begin{document}$ u(x,t) $\end{document}</tex-math></inline-formula> is analytic in time at <inline-formula><tex-math id="M9">\begin{document}$ t = 0 $\end{document}</tex-math></inline-formula>. Applying this method, we also obtain a necessary and sufficient condition for the solvability of the backward equations, which is ill-posed in general.</p><p style='text-indent:20px;'>For the nonlinear heat equation with power nonlinearity of order <inline-formula><tex-math id="M10">\begin{document}$ p $\end{document}</tex-math></inline-formula>, we prove that a solution is analytic in time <inline-formula><tex-math id="M11">\begin{document}$ t\in (0,1] $\end{document}</tex-math></inline-formula> if it is bounded in <inline-formula><tex-math id="M12">\begin{document}$ \mathrm{M}\times[0,1] $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M13">\begin{document}$ p $\end{document}</tex-math></inline-formula> is a positive integer. In addition, we investigate the case when <inline-formula><tex-math id="M14">\begin{document}$ p $\end{document}</tex-math></inline-formula> is a rational number with a stronger assumption <inline-formula><tex-math id="M15">\begin{document}$ 0&lt;C_3 \leq |u(x,t)| \leq C_4 $\end{document}</tex-math></inline-formula>. It is also shown that a solution may not be analytic in time if it is allowed to be <inline-formula><tex-math id="M16">\begin{document}$ 0 $\end{document}</tex-math></inline-formula>. As a lemma, we obtain an estimate of <inline-formula><tex-math id="M17">\begin{document}$ \partial_t^k \Gamma(x,t;y) $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M18">\begin{document}$ \Gamma(x,t;y) $\end{document}</tex-math></inline-formula> is the heat kernel on a manifold, with an explicit estimation of the coefficients.</p><p style='text-indent:20px;'>An interesting point is that a solution may be analytic in time even if it is not smooth in the space variable <inline-formula><tex-math id="M19">\begin{document}$ x $\end{document}</tex-math></inline-formula>, implying that the analyticity of space and time can be independent. Besides, for general manifolds, space analyticity may not hold since it requires certain bounds on curvature and its derivatives.</p>

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 [6,8,26] that solutions to the heat equation in the whole space, or in the 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.

The existence of smooth but nowhere analytic functions is well-known (du Bois-Reymond [Math. Ann. 21 (1883), no. 1, pp. 109–117]). However, smooth solutions to the heat equation are usually analytic in the space variable. It is also well-known (Kowalevsky [Crelle 80 (1875), pp. 1–32]) that a solution to the heat equation may not be time-analytic at t = 0 t=0 even if the initial function is real analytic. Recently, it was shown by Dong and Pan [J  Math. Fluid Mech. 22 (2020), no. 4, Paper No. 53]; Dong and Zhang [J. Funct. Anal. 279 (2020), no. 4, Paper No. 108563]; Zhang [Proc. Amer. Math. Soc. 148 (2020), no. 4, pp. 1665–1670] that solutions to the heat equation in the whole space, or in the half space with zero boundary value, are analytic in time under an essentially optimal growth condition. 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 \delta >0 , we find a solution to the heat equation on the whole plane, with exponential growth of order 2 + δ 2+\delta , which is nowhere analytic in time.