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I'm reading Evan's Partial Differential Equations. In the proof of the Theorem 1(iii) (Initial Value Problem of the Heat Equation on Page 48), the last step states $$\frac{C}{t^{n/2}}\int_{\mathbb{R}^n-B(x^0,\delta)}e^{-\frac{|y-x^0|^2}{16t}}dy=\frac{C}{t^{n/2}}\int_\delta^\infty e^{-\frac{r^2}{16t}}r^{n-1}dr$$ I think it's just a change of variable $r=|y-x^0|$. If so, $dr=\frac{y-x^0}{r}dy$. But I don't know how to proceed from here, and I don't see where that $r^{n-1}$ comes from. Can anyone help? Thanks in advance!

IntegralLover
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    The $r^{n-1}$ shouldn’t be in the exponential, it should be multiplying out front. THis is due to the Jacobian factor from the change of variables to spherical coordinates (also, there should be an extra factor of the surface area of the unit sphere, but perhaps Evans just absorbed it into the constant $C$). – peek-a-boo Oct 03 '23 at 19:44
  • Yea you are right. I've corrected it. But it seems that it still has some problem, as Evans edited it to $C\int_{\mathbb{R}^n-B(0,\frac{\delta}{\sqrt{t}})}e^{-\frac{|z|^2}{16}}dz$ in the second edition. – IntegralLover Oct 03 '23 at 19:50
  • You want to evaluate the RHS? Try u-sub then IBP assuming $n\geq 2$. For $n=1$ this is just a Gaussian. – Mr. Brown Oct 03 '23 at 20:15

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