# Shock formation in $u_{tt} = (1+u_x)^2 u_{xx}$

In Sergiu Klainerman's paper "Global Existence for Nonlinear Wave Equations", we are given the PDE $$u_{tt} = (1+u_x)^2 u_{xx}$$ as an example of a wave equation which exhibits blowup. Klainerman gives a specific example of this: for smooth $$H\in C_0^\infty(\mathbb R)$$, he claims that the initial conditions $$u(0,x) = H(x)$$ and $$u_t(0,x) = -H'(x)-\frac12 H'(x)^2$$ lead to a shock at time $$t=-1/h$$ and at $$x=\xi$$, where $$h = \min H''$$ and $$h = H''(\xi)$$.

This seems like an argument based on characteristics, but I'm not familiar enough with characteristics for second order PDEs to see how to get here.

As described in this post, the following simple wave solution can be derived using the method of characteristics: $$u = H\big(x - (1+\theta) t\big) +\tfrac12 \theta^2 t \, ,$$ where $$\theta=u_x$$ satisfies the implicit equation $$\theta = H'(x - (1+\theta) t)$$. The initial time derivative is necessarily of the desired form $$u_t(0,x) = -\left(H'(x)+\tfrac12 H'(x)^2\right) .$$ and the breaking time reads $$t=- 1/\inf H''$$.