Behavior of a solution to the heat equation at infinite time

Question

Could I get a hint as to how I find the limit as $t \to \infty$ of this solution to the heat equation:

$$u(x,t) = \int_{-\infty}^{\infty} \frac{1}{\sqrt{4\pi t}} e^{\frac{-|x-y|^2}{4t}} e^{|y|} dy$$

I've seen this related question but I'm not sure if it helps me because $e^{|y|}$ is not a bounded at infinity:

Long time behavior heat equation on infinite line

Answer 1

The integral is not convergent as $t\to\infty$.

See below the explicit expression of $u(x,t)$ which tends to infinity when $t\to\infty$.