Similarity variables

Fix a time $T$ , put $\tau=T-t>0$ , and choose real exponents $a,b_1,\ldots,b_d$ . Similarity variables for a field $u(t,x)$ are

y_i=x_i\tau^{-b_i},\qquad s=-\log\tau,\qquad U(s,y)=\tau^a u(T-\tau,D_\tau y),

where $D_\tau$ is the corresponding anisotropic dilation. The field need not be invariant under this change of variables.

Derivative formulas

The chain rule for $u=\tau^{-a}U(s,y)$ gives

\partial_tu=\tau^{-a-1}\left(aU+\partial_sU+\sum_i b_i y_i\partial_{y_i}U\right), \qquad \partial_{x_i}u=\tau^{-a-b_i}\partial_{y_i}U.

These formulas describe the coordinate change for any exponents. Whether they make a particular PDE autonomous requires substituting into that equation.