Questions on "Partial Differential Equations", as opposed to "ordinary differential equations".

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The path of the shock

Here I am using the shock speed to work out the path the shock takes. I don't understand why we cannot take the value of $u_{-}$ at $t=1/u_0$ i.e $u_{-}=u_0$. and calculate the speed of the ...
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Does $\operatorname{div}\left(\nabla G +xG\right)=0\Longleftrightarrow \nabla G +xG=0$?

Let $G$ be a function of $\mathcal{C}^2(\textbf{R}^d$;$\textbf{R}^*_+)$ such that $G \in \operatorname{L}^1(\textbf{R}^d)$. I read on the Internet that one has the following equivalence ...