I have recently asked a similar question but did not formulate it well. Let us consider a weak solution $u(y,s)$ to the heat equation $$\partial_t u - \Delta u=0$$ in some space-time cylinder $B_r(x)\times(t-\alpha r^2,t+\alpha r^2)$ for some $x\in\mathbb{R}^n, t>0$ and $\alpha>0$ such that $(t-\alpha r^2)>0$. In a paper it is claimed that, after re-scaling to the unit ball and unit interval, the scaled function $v(z,q) = \frac{1}{\alpha}u(x+rz,t+r^2q)$ for $(z,q)\in B_1(0)\times(-1,1)$ is again a weak solution to the heat equation, but now in the scaled domain $B_1(0)\times(-1,1)$.
I can see what the scaling does and I would agree that in the classical setting, i.e. if $u$ was a classical solution, then also $v$ would be a classical solution. However, in the weak formulation I cannot verify that this really holds as for the time derivative term (as in the weak form there is no time derivative on $u$ anymore but only on the test function) I only get the factor $\frac{1}{\alpha}$ whereas for the Laplacian term I obtain the factor $\frac{r}{\alpha}$ and thus not a common factor I could factorize out of the integral. Did I just make a mistake in my calculation or am I misunderstanding anything (or is the paper wrong which i highly doubt)? I hope the question is stated clearly this time. If not, please let me know! Thanks for any help in advance!