The Cauchy problem I'm dealing with leads to the following Hopf-Lax function:
$$u(t,x) = \inf_{y\in \mathbb{R}} \left\{ 2|y|^2 + \frac{|x-y|^2}{2t} \right\},$$
but I don't know how to explicitly calculate this infimum. Could any solutions be step by step, please.
You don't really need the absolute values if everything is real, so you're looking for the minimum of a quadratic in $y$. As long as $2 + 1/(2t) > 0$, the $y^2$ term in this quadratic is positive, so the critical point is a minimum. You can find it by the usual calculus method: take the derivative, set it to $0$, and solve. But if $2 + 1/(2t) < 0$ (or $=0$ with $x \ne 0$), the infimum will be $-\infty$.