For some $\alpha>1$, define a function $\varphi$ on $\mathbb{R}_{+}$ by $$\varphi(t):=\begin{cases}e^{-t^{-\alpha}}, & {t>0}\\ 0, & {t\leq 0}\end{cases}$$
It is well-known that $\varphi$ is $C^{\infty}(\overline{\mathbb{R}}_{+})$ and $\varphi^{(k)}(0)=0$ for all integers $k\geq 0$. Tychonoff solution of heat equation can be defined as $u:\mathbb{R}\times\mathbb{R}_{+}\rightarrow\mathbb{R}$ defined by
$$u(x,t):=\sum_{k=0}^{\infty}\dfrac{\varphi^{(k)}(t)}{(2k)!}x^{2k}, \qquad (x,t)\in\mathbb{R}\times\mathbb{R}_{+}$$
which belongs to $C^{\infty}(\mathbb{R}_{+}\times\mathbb{R})$.
My question is why $u(x,t)$ does not satisfy the growth condition $$ |u(x,t)|\leq Ae^{a|x|^2} $$ near $t=0$.
Can we prove it by calculating directly without using the uniqueness theorem for the Cauchy problem?
