# On the propagation of singularities in PDE

This question might be a little generic, but i wanted to get some idea on the concept of propagation of singularities in PDE. Searching the internet i only found very complicated things about the subject, so if someone could give an idea, more or less precise of this and it is related to hypoellipticity i would be grateful! Also providing some examples for classical heat, wave, schrodinger equations. Thanks for any help.

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The simplest equation for which singularities propagate is the 1d wave equation $u_{tt}=c^2u_{xx}$: see the nice exposition by Jiří Lebl with some plots of solution.
@balestrav Yes, as the plots in Lebl's book show, the solution that was nonsmooth at $x=0, t=0$ will be nonsmooth at $x=\pm ct$, $t=0$: so, the singularity propagates left and right at speed $c$. In this example it's just the lack of derivative, but a jump discontinuity (wave front) would behave in the same way. –  user31373 Jun 27 '12 at 21:15