Consider the heat equation $\partial_t u - \Delta u = 0, u(0, x) = u_0$ on a complete (non-compact) Riemannian manifold $M$, may be even $\mathbb{R}^n$. I was wondering, what are some known sufficient conditions under which the solution to the heat equation becomes instantaneously real analytic in time? This is mainly a reference request: the only reference I know is K. Yosida's paper ``An Abstract Analyticity in Time for Solutions of a Diffusion Equation'' (available here). Also, with regard to the main result of Yosida's paper, I have the further question: does one need to have a spatial bound on the solution (like $L^\infty$-bound on $u$, or $u \in L^2$, as in Yosida)? The reason I am asking about $L^\infty$ bound on $u$ is, in Tychonoff's example, (see this MO post), $u$ grows rapidly at infinity, and the solution constructed turns out to be not real analytic.
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2$\begingroup$ I guess the manifold $M$ had better be real analytic (with real analytic transition functions); otherwise your question is not even well defined. $\endgroup$– Fan ZhengCommented Oct 4, 2015 at 0:01
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1$\begingroup$ @FanZheng Aren't all smooth manifolds real analytic with the two structures being compatible? I thought this was a result of Whitney. $\endgroup$– SMSCommented Oct 4, 2015 at 12:00
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1$\begingroup$ @FanZheng Besides, even if they weren't, I don't see clearly how that affects this issue. I only want real analyticity with respect to the variable $t$. May be I am not thinking clearly. $\endgroup$– SMSCommented Oct 4, 2015 at 12:03
3 Answers
In the case of $\mathbb R^n$, the analyticity-in-time of the solution, with $u_0\in\mathcal S'$ (tempered distributions), stems from that of $e^{-(t+i\tau)|\xi|^2}$ which implies that of $u(t+i\tau,x)=\mathcal F^{-1}_{\xi\to x}[\hat u_0(\xi)e^{-(t+i\tau)|\xi|^2}]$ as a function of the complex variable $t+i\tau$, $t>0$.
For manifolds, I don't know, but the question makes sense, even if the metric is not analytic (since we're discussing analyticity in time).
Perhaps I am overlooking something, but a positive answer to your question looks rather straightforward to me, provided the Laplace-Beltrami operator on the manifold is defined suitably -- i.e., weakly.
Consider the closed quadratic form $a(u):=\int_M |\nabla u|^2 \ d\sigma$ for $u\in H^1(M)$. The associated operator is $\Delta$ and it generates an analytic semigroup by the general theory of linear semigroups. Voilà.
(Regularity in space - and in particular the possibility to go back from a weak formulation of the parabolic equation to the strong one -- is a more subtle issue, due to the fact that analytic semigroups enjoy very good smoothness with respect to the domains of their generators' powers, hence typically in terms of Sobolev spaces; but on a singular manifold Sobolev imbeddings may be hard to prove.)
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$\begingroup$ The point is that on e.g. $\mathbb{R}^n$ the Tychonoff solution exists, and that is not analytic in time. So it is not quite as straightforward as "voila". $\endgroup$ Commented Sep 13, 2019 at 0:22
There seems to be an answer in the paper: https://arxiv.org/abs/1907.01687 Time analyticity for the heat equation and Navier-Stokes equations
