Denote $\Gamma(x,t)$ the fundamental solution of the heat equation form thwe integral. By the theorem of L. Schwarz for any $\mu\in S'(S)$ there is a number $m\in \mathbb N$ and $C>0$ such that $$\|u(x,t)\|=|(\mu,\Gamma(t,\cdot))|\le C\|\Gamma(t,\cdot))\|_m,$$ where $ \|\varphi\|_m=$
$\sup_{\alpha \le m,\ x \in \mathbb R}(1+|x|)^m |\partial^\alpha \varphi(x)|. $
It is straightforward to obtain, as it is said in the comments, that the $u$ increases at most polynomially. Namely, it is known that $$|\partial_x^k \Gamma(t,x)|\le C_k t^{-(k+1)/2}e^{-c_k x^2/t}.\ $$ From here it is easy to get $$|x^m \partial_x^k \Gamma(t,x)|\le C_{k,m} t^{(m-k-1)/2}e^{-c_{k,m} x^2/t},$$ since $|y|^m e^{-c y} \le C_m$. Putting $y=x^2/t\ $ we have $|x|^m e^{-c x^2/t}\le t^{m/2}e^{-c x^2/2t}$.