why is the ratio $\frac{r^2}{t}$ important for solving the heat equation?

Question

I understand how $u(\lambda x, \lambda^2t)$ is also a solution if $u$ solves the heat equation but I can't link this to that specific ratio.

can someone explain the relationship between the two ?

Answer 1

You can find that ratio by a scaling argument. Let $v(x,t):=u(ax,bt)$ with $a,\ b>0$. We will impose that $v$ solves the heat equation and this will give us a relation on $a,\ b$. We set also $y=ax$, $s=bt$.
We have
$$v_t(x,t)=b u_s(y,s),$$ $$\partial_{x_i}v(x,t)=a\partial_{y_i} u(y,s),$$ $$\partial^2_{x_i}v(x,t)=a^2\partial^2_{y_i} u(y,s).$$ Thus $$v_t(x,t)-\Delta v(x,t)=bu_s(y,s)-a^2\Delta u(y,s)$$ and $v$ solves the heat equation if and only if $a^2=b$.