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6 votes

Nontrivial invariant transformations for heat equations

Yes, how about the Appell transform, see here. $$v(t,x)\mapsto\Gamma(t,x)v(-\frac1t,\frac xt)$$ where $\Gamma$ is the heat kernel. Truly nontrivial if you ask me. Of course, this inverts time and not ...
Funktorality's user avatar
3 votes
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Nontrivial invariant transformations for heat equations

At least when $n=1$, there are no nontrivial transformations of this kind. Indeed, suppose that $v(t,x)=u(\tau(t,x),\xi(t,x))$, where $u_t=u_{xx}$. Then $u_{tx}=u_{xxx}$ and $u_{tt}=u_{xxxx}$, so that ...
Iosif Pinelis's user avatar

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