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I believe that the following result is true, but am having a hard time finding a suitable reference to convince myself:

Suppose $X:\mathbb{R}^n\to \mathbb{R}^n$ is analytic. If the smooth function $u:(0,\infty)\times \mathbb{R}^n\to \mathbb{R}$ solves the parabolic equation

$$\frac{\partial u}{\partial t}=\Delta u+ X\cdot \nabla u,$$ then for each $t>0$, the mapping $x\mapsto u(t,x)$ is analytic.

The elliptic version of this result is frequently cited as "well-known" in the literature, and in fact, there is this result for elliptic non-linear systems as well: https://www.jstor.org/stable/2372830.

The parabolic result is for the regular heat equation ($X=0$) is discussed here: Space analyticity of solution to heat equation. However, the proof it refers to appears to lean quite heavily on an explicit expression for the heat kernel/fundamental solution, which might not be available if $X\neq 0$.

Thanks in advance.

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  • $\begingroup$ Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. $\endgroup$
    – Community Bot
    Feb 27 at 22:51

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I think the result you’re looking for is given in:

Classes of solutions of linear systems of partial differential equations of parabolic type by Avner Friedman.

Duke Math. J. 24 (3), 433-442, (September 1957) DOI: 10.1215/S0012-7094-57-02450-X

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    $\begingroup$ Thanks! That does the job quite nicely I believe. $\endgroup$ Feb 28 at 5:37

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