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We study the time analyticity of ancient solutions to heat equations on graphs. Analogous to Dong and Zhang [DZ19], we prove the time analyticity of ancient solutions on graphs under some sharp growth condition.
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Mean curvature flow for isoparametric submanifolds in Euclidean spaces and spheres was studied by the authors in [LT]. In this paper, we will show that all these solutions are ancient solutions. We also discuss rigidity of ancient mean curvature flows for hypersurfaces in spheres and its relation to the Cherns conjecture on the norm of the second fundamental forms of minimal hypersurfaces in spheres.
By studying the heat semigroup, we prove Li-Yau type estimates for bounded and positive solutions of the heat equation on graphs, under the assumption of the curvature-dimension inequality $CDE(n,0)$, which can be consider as a notion of curvature for graphs. Furthermore, we derive that if a graph has non-negative curvature then it has the volume doubling property, from this we can prove the Gaussian estimate for heat kernel, and then Poincare inequality and Harnack inequality. As a consequence, we obtain that the dimension of space of harmonic functions on graphs with polynomial growth is finite, which original is a conjecture of Yau on Riemannian manifold proved by Colding and Minicozzi. Under the assumption of positive curvature on graphs, we derive the Bonnet-Myers type theorem that the diameter of graphs is finite and bounded above in terms of the positive curvature by proving some Log Sobolev inequalities.
In this paper, we investigate pointwise time analyticity of solutions to fractional heat equations in the settings of $mathbb{R}^d$ and a complete Riemannian manifold $mathrm{M}$. On one hand, in $mathbb{R}^d$, we prove that any solution $u=u(t,x)$ to $u_t(t,x)-mathrm{L}_alpha^{kappa} u(t,x)=0$, where $mathrm{L}_alpha^{kappa}$ is a nonlocal operator of order $alpha$, is time analytic in $(0,1]$ if $u$ satisfies the growth condition $|u(t,x)|leq C(1+|x|)^{alpha-epsilon}$ for any $(t,x)in (0,1]times mathbb{R}^d$ and $epsilonin(0,alpha)$. We also obtain pointwise estimates for $partial_t^kp_alpha(t,x;y)$, where $p_alpha(t,x;y)$ 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 $mathrm{M}$, 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 $mathrm{M}$ satisfies the Poincare inequality and the volume doubling condition. Moreover, we also study the time and space derivatives of the fractional heat kernel in $mathbb{R}^d$ using the method of Fourier transform and contour integrals. We find that when $alphain (0,1]$, the fractional heat kernel is time analytic at $t=0$ when $x eq 0$, 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 [11] which deal with local equations.
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.
In this article, we study certain type of boundary behaviour of positive solutions of the heat equation on the upper half-space of $R^{n+1}$. We prove that the existence of the parabolic limit of a positive solution of the heat equation at a point in the boundary is equivalent to the existence of the strong derivative of the boundary measure of the solution at that point. Moreover, the parabolic limit and the strong derivative are equal.
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