Assume $u(x,t)$ is solution of $$ \partial_t u(x,t) =\Delta_x u(x,t) $$ for $t\in [a,b]$. Seemly, there is a method (called parabolic rescaling and time shift) can make a solution $\hat u(\hat x, \hat t)$ such that $$ \partial_\hat t \hat u(\hat x,\hat t) =\Delta_\hat x \hat u(\hat x,\hat t) $$ for $\hat t\in [0,1]$.
What I try: let $\hat t = \frac{t-a}{b-a}$ and $\hat u (x,\hat t)= u(x, a+\hat t (b-a))=u(x,t)$, then I have $$ \partial_{\hat t} \hat u(x,\hat t) = (b-a)\Delta_x \hat u (x, \hat t) $$ Then, I don't know how to do it.