some point $y_0$ will shock in finite time. Let us assume that $w_0$ has a global minimum slope. By temporal rescaling and translation invariance, without loss of generality, we may assume the global minimum slope is $-1$, occurring at $y=0$. Let us take the initial time to be $t=-1$. By Galilean symmetry, without loss of generality, we may further assume $w_0(0)=0$, then by methods of characteristics that the solution $w$ will shock at $(y,t)=(0,0)$.

If in addition $w'''_0(0) =\nu> 0$, then the solution $w$ will convergence asymptotically at the blow up to a self-similar profile ${\overline}W_1$; in particular, we have $$\begin{aligned}
 \label{limit:1}
\lim_{t\rightarrow 0} (-t)^{-\frac12}w(x(-t)^{-\frac32},t)= \left(\frac{\nu}{6}\right)^{-\frac12}{\overline}{W}_1\left(\left(\frac{\nu}{6}\right)^{\frac12}x\right)\,,\end{aligned}$$ for any $x\in\mathbb R$, where $$\begin{aligned}
 \label{explicit:1}
 {\overline}{W}_1(x) = \left(- \frac x2 + \left(\frac{1}{27} + \frac{x^2}{4}\right)^{\frac 12}\right)^{\frac 13} - \left( \frac x2 + \left(\frac{1}{27} + \frac{x^2}{4}\right)^{\frac 12} \right)^{\frac 13} \,.\end{aligned}$$

Note one can fix $\nu$ by making use of of the spacial rescaling invariance of Burgers’ equation.

The shock profile is stable in the sense that given any initial data in a suitably small $C^4$ neighborhood of $w_0$, the resulting solution will satisfy modulo the invariances of Burgers’ equation. The profile ${\overline}W_1$ (together its $\nu$ rescaling given on the right hand side of ) 