Given the following PDE: $$∂p/∂t+p*∂p/∂x=0$$ $$p=p(x,t)$$ $$p(x,0)=e^{-x^2}$$
It is requested to find its solution, show that it is not well behaved by verifying what happens to $∂p/∂t$ and $∂p/∂x$ when $t*=(e)^{x_0²}/2x_0$ and to show that $t^{*}=(e/2)^{1/2}$ is the first $t^{*}$ where those derivatives will be singular - and is the moment when the shock occurs.
I found, for the solution, that $p(x,t)=(e)^{-x_0²}$ for all $(x,t)$ such that $x=xo+(e)^-{x_0²}*t$ (implicit solution). I also found, for $∂p/∂t$ and $∂p/∂x$, $1/(t^{*}*e^{x_o²}(1-t*t^{*}))$ and $-1/t^{*}$, respectively. I then derived the $t^{*}$ and put it equal to zero to find the critical points and got the value requested, $(e/2)^{1/2}$.
The problem: I can't explain exactly - in good terms - why that is the point where the derivatives are SINGULAR and why that is the moment the shock occurs. I believe shock means shock of solutions, as regarding to the blowup time.