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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.

Dominic Wynter
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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 $t=- 1/\inf H''$.

EditPiAf
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