Growth of Tychonov's Counterexample for Heat Equation Uniqueness

Define a function $\varphi$ on $\mathbb{R}_{+}$ by $$\varphi(t):=\begin{cases}e^{-1/t^{2}}, & {t>0}\\ 0, & {t\leq 0}\end{cases}\tag{1}$$

It is well-known that $\varphi$ is $C^{\infty}(\overline{\mathbb{R}}_{+})$ and $\varphi^{(k)}(0)=0$ for all integers $k\geq 0$. Tychonoff famously showed that the function $u:\mathbb{R}_{+}\times\mathbb{R}\rightarrow\mathbb{R}$ defined by

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

belongs to $C^{\infty}(\mathbb{R}_{+}\times\mathbb{R})$ and satisfies the Cauchy problem

$$\begin{cases}\partial_{t}u-\Delta u=0 & {(t,x)\in\mathbb{R}_{+}\times\mathbb{R}},\\ u(0,\cdot)=0 & {}\end{cases}\tag{3}$$

Question. Motivated by this question, I am trying to show that for $t>0$ fixed, $u(t,\cdot)$ does not define a tempered distribution. I am looking for a "lower bound" on the growth of $u$ for $t>0$ fixed which I can use to construct a sequence of test functions $\varphi_{m}\in\mathcal{S}(\mathbb{R}^{n})$ which tend to zero in the Schwartz topology but

$$\left|\langle{u(t,\cdot),\varphi_{m}}\rangle\right|=\left|\int_{\mathbb{R}^{n}}u(t,x)\varphi_{m}(x)dx\right|\geq c,\quad\forall m\in\mathbb{N}\tag{4}$$

for some $c>0$.

I know that the function $u$ does not satisfy the growth condition

$$\sup_{0\leq t\leq T}\left|u(x,t)\right|\leq Ae^{c\left|x\right|^{2}},\quad\forall x\in\mathbb{R}^{n}\tag{5}$$

where $T>0$ is fixed and $A,c>0$ are constants depending on $T$; however, I fail to see how this helps in the task described above.

Edit: Einar Rødland has presented some graphical evidence to suggest that for fixed $t>0$, $u(t,\cdot)$ is "well-behaved" and defines a tempered distribution. I am seeking a proof of disproof of this conjecture. Note that a "wild solution" can be "well-behaved" for $t>0$ fixed. For example, in this paper, the authors present an example of nonuniqueness for the Cauchy problem which is continuous on $\mathbb{R}\times [0,\infty)$ and satisfies

$$\left|u(x,t)\right|\leq C e^{\epsilon/t},\qquad (x,t)\in\mathbb{R}\times\mathbb{R}_{+}$$

where $C=C(\epsilon)$, for any $\epsilon>0$. Also, at the end of the paper the authors remark that all other (i.e. besides theirs) nonuniqueness solutions are unbounded in $x$, which would seem to contradict Einar's suggestion.

Solution 1:

Here's a summary of, and extending remarks and explanations to, the comments. Note that my description of $u(t,x)$ is based on experimental observations, not formal proofs.

I plotted the sum for $u(t,x)$ for a few values of $t$: within a moderate range for $x$, it would converge, although I did need extended accuracy (used Maple for this). My understanding is largely based on what these plots looked like.

Basically, for $t\le0$ we have $u(t,x)=0$. But then, for $t>0$, the $u(t,x)$ looks like a wave packet moving in from infinity towards the origin. For a given $t>0$, beyond the region containing the wave packet, the function levels off quickly to $1$ as $x$ increases. As $t$ increases, this wave moves towards the origin and dampens out, with $u$ eventually converging to constant $1$.

However, if we move back in time towards $t=0^+$, the wave pattern is more violent, spread out, and further away from the origin: i.e., higher amplitude and wave spreads further out to higher values of $x$. As we lower $t$ towards 0, and the wave packet is further away from the origin, $u(t,x)$ converges towards $0$ for any fixed $x$.

Here's a plot showing $y=u/[(1+u^2)/2]^{1/4}$ (had to compress the scale to enable large and small amplitudes to show), for $t=1$, $1/2$, $1/3$, $1/4$, $1/5$, $1/6$, and $1/7$, with $t=1$ having the smoothest curve and $t=1/7$ the most variable.

Thus, for any given $t>0$, we have a nice solution with no exotic behaviour. Moreover, for $t\le0$, we have $u(t,x)=0$ which means the solution for $t>0$ does not arise from prior conditions. The extreme behaviour is only apparent as $t\rightarrow 0^+$. Thus, for any given $t>0$ the solution is a tempered distribution (and should also be a Schwartz function), but $u(t,x)$ is wildly unbounded as $t\rightarrow 0^+$.

NB: The solution of $u$ expressed from $\phi(t)$ works for any $C^\infty$ function, as far as I can tell. What makes $\phi(t)=e^{-1/t^2}$ for $t>0$, $\phi(t)=0$ for $t\le0$, special is that all derivatives are zero for $t\le0$ allowing $u(t,x)$ to be defined as continuous in both $t$ and $x$, while identically zero for $t\le0$.