= Solution
Let $\widehat\sigma(x,t)=\sigma(e^x,t)$, $a(x,t)=\widehat\sigma^2/2$ and $b(x,t)=r-\widehat\sigma^2/2$. The chain rule gives $SV_S=V_x$ and $S^2V_{SS}=V_{xx}-V_x$. Also $V=e^{-r(T-t)}u$, so $V_t=e^{-r(T-t)}(u_t+ru)$. The killing term cancels, leaving
$$
\boxed{u_t+a(x,t)u_{xx}+b(x,t)u_x=0,\qquad u(x,T)=(e^x-K)^+.}
$$
This is the <explicit log-price scheme for local-volatility pricing> equation. Under the <risk-neutral measure>, the corresponding log-price <Itô formula> gives
$$
\boxed{dx_t=\left(r-\frac12\sigma(e^{x_t},t)^2\right)dt+\sigma(e^{x_t},t)dW_t^Q.}
$$
The logarithmic coordinate removes the <stock>-price factors from the diffusion coefficient, but retains the spatial variation of <local volatility>.
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