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We say that a function $g\in C^{\infty}(\mathbb{R}^{n})$ is of Gevrey class $G^{\sigma}(\mathbb{R}^{n})$ if for each compact $K\subset\mathbb{R}^{n}$, there exist $C,R>0$ such that

$$\sup_{x\in K}\left|D^{\alpha}g(x)\right|\leq CR^{k}k^{\sigma k}, \quad \left|\alpha\right|=k$$

For $\sigma>1$, one can show that there exist compactly supported elements in $G^{\sigma}(\mathbb{R}^{n})$, which are not identically zero.

Exercise 2, Section 5 of M.E. Taylor, Partial Differential Equations I (2nd Edition) asks the reader to show that if $g\in G^{\sigma}(\mathbb{R}^{n})$, for $1<\sigma<2$, that the series

$$u(t,x):=\sum_{k=0}^{\infty}\dfrac{x^{2k}}{(2k)!}g^{(k)}(t), \qquad (t,x)\in\mathbb{R}\times\mathbb{R}$$

solves the heat equation with initial datum $u(t,0)=g(t)$. It then asks the reader to using a compactly supported Gevrey class function to find a nontrivial solution to the heat equation supported in the strip $a\leq t\leq b$. My issue is not with either of these tasks, but rather a comment made by the author concerning the growth of such a solution.

Question. At the bottom of pg. 245, the author remarks that for "fixed $t>0$, such solutions blow up too fast to belong to $\mathcal{S}'(\mathbb{R}^{n})$." How does one obtain a lower bound on the growth of $u(t,\cdot)$ to prove this?

My original idea was to suppose $u(t,\cdot)$ is a tempered distribution for some $t_{0}>0$ such that $u(t_{0},\cdot)\not\equiv 0$. If I could show that the map $t\mapsto u(t,\cdot)$ belonged to $C^{1}([t_{0},\infty),\mathcal{S}'(\mathbb{R}^{n}))$. We would then arrive at a contradiction by applying the uniqueness result for this class with initial datum $u(t_{0},\cdot)$, since $u(t,x)$ vanishes identically for all $t$ sufficiently large. An another idea is to somehow use the structure theorem for tempered distributions, but the problem here is that I do not know what $\left\{g^{(k)}(t)\right\}_{k}$ looks like for arbitrary $a\leq t\leq b$.

Motivation. I am so interested in this proving this property because it seems that examples of nonuniqueness for the heat equation Cauchy problem can exhibit different behavior for $t>0$ fixed.

EDIT: So Ian made a remark that made me think the author's claim is not true as stated. First, it's trivially not true for $t\notin [a,b]$ since $u(t,x)\equiv 0$. Second, couldn't we construct $g$ to be a nonzero constant on some small subinterval of $[a,b]$, in which case $u(t,x)$ is polynomial for some times?

EDIT2: I think the original question may have been unclear in that I do not know that for some $t>0$ fixed, $\left|u(t,x)\right|$ grows very rapidly (e.g. dominates $e^{c\left|x\right|^{2}}$ for some $c>0$) as $\left|x\right|\rightarrow\infty$; this is for what I am asking.

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Matt Rosenzweig
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    I have seen this proven more or less by contraposition: among functions which grow slower than $e^{c t^2}$ for all $c>0$, the solution to the heat equation is unique. Functions which grow faster than $e^{c t^2}$ are not in $\mathcal{S}'$; one proof is to note that they cannot be integrated against $e^{-ct^2/2}$. So then you can proceed by showing that your function grows faster than $e^{ct^2}$. I think Evans covers this. – Ian May 22 '15 at 14:54
  • @Ian: The solution is unique on $\mathbb{R}\times [0,T]$ in the class satisfying the growth estimate $\sup_{0\leq t\leq T}\left|u(x,t)\right|\leq Ae^{cx^{2}}$ for some constants $A,c>0$. This the growth condition originally due to Tychonoff, which I think you are thinking of and Evans covers. – Matt Rosenzweig May 22 '15 at 14:58
  • Er, yes, that's what I was going for, sorry. So you can prove your result by showing that the growth estimate fails. Then the result will not be in $\mathcal{S}'(\mathbb{R}^n)$ for some times. – Ian May 22 '15 at 15:00
  • @Ian: If my solution fails this growth estimate, that just means there are times $t_{j}$ such that $\left|u(t_{j},x)\right|\geq Ae^{cx^{2}}$ on some interval, but I don't know how large the interval is. So I don't see how this estimate helps me. – Matt Rosenzweig May 22 '15 at 15:03
  • @Ian: If you can somehow make that argument work, please do share your solution. I am skeptical, though, because there is an example of nonuniqueness, though not compactly supported in time, which is a tempered distribution for all $t>0$ fixed. – Matt Rosenzweig May 22 '15 at 16:11
  • @Ian: I have edited the question to include a counterexample to your claim that "[f]unctions which grow faster than $e^{ct^{2}}$ are not in $\mathcal{S}'$". So it appears that something more than the argument you suggested is needed. – Matt Rosenzweig May 23 '15 at 22:48
  • Er...as stated your inequality makes no sense, because after taking the supremum there is no longer an $x$ dependence, so the supremum can't be dominated by a function of $x$. Do you mean to say that your example is not globally dominated by any $e^{c x^2}$? If so, then yes, you're right. My point was actually that if a function dominates $e^{cx^2}$ for some $c$ then it cannot form a tempered distribution. I think anything in $L^p \setminus L^\infty$ will furnish an example of the sort you intend. – Ian May 23 '15 at 23:28
  • @Ian: Yes, I meant dominated by a function of $x$--my bad. Ah, I misunderstood your point. Am I missing something then in your original two comments? That the solution $u$ fails the $\sup_{0\leq t\leq T}\left|u(x,t)\right|\leq Ae^{c x^{2}}$ doesn't imply that for some time $t$, $u(t,\cdot)$ dominates $e^{cx^{2}}$ for some $c>0$. – Matt Rosenzweig May 24 '15 at 00:02
  • I have a paperback edition, and cannot find the remark. Can you give a section reference? – timur May 30 '15 at 20:58
  • @Timur: A section reference for what? The quote from Taylor is section 5. For Evans, the relevant section is 2.3.3. – Matt Rosenzweig May 30 '15 at 21:04
  • Ok, I found it. What prevents you from applying the uniqueness part of Proposition 5.1 to produce contradiction? Smoothness in time? – timur May 30 '15 at 21:20
  • @timur: The uniqueness statement is only for the class $C^{\infty}(\mathbb{R}^{+},\mathcal{S}(\mathbb{R}^{n}))$. We run into problems as $t\downarrow 0$, namely the solution is very large on some interval of space. In the paper S.Y. Chung and D. Kim, "An Example of Nonuniqueness of the Cauchy Problem for the Heat Equation", the authors construct an example which is uniformly bounded with respect to x and in particular belongs to S'(R^n) for t>0 fixed. – Matt Rosenzweig May 30 '15 at 21:28
  • Isn't running into problems and solution being large exactly what we want? – timur May 30 '15 at 21:55
  • @Timur: Not if the information is just qualitative. All we know is that $u$ fails the growth condition stated above. I don't know for how large an interval of space the solution is for fixed $t>0$. This was discussed in the comments above. Note that there is an error in my previous comment. The uniqueness class is $C^{\infty}(\overline{\mathbb{R}^{+}},\mathcal{S}'(\mathbb{R}^{n}))$. – Matt Rosenzweig May 30 '15 at 22:07

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