On the formation of shocks for quasilinear wave equations

Understand

The paper is devoted to the study of shock formation of the 3-dimensional quasilinear wave equation \begin{equation}\label{Main Equation} - \big(1+3G^{\prime\prime}(0) (\partial_t\phi)^2\big)\partial^2_t \phi +\Delta\phi=0,\tag{\textbf{$\star$}} \end{equation} where $G^{\prime\prime}(0)$ is a non-zero constant.

• We will exhibit a family of smooth initial data and show that the foliation of the incoming characteristic hypersurfaces collapses.
• Similar to 1-dimensional conservational laws, we refer this specific type breakdown of smooth solutions as shock formation.
• Since $(\star)$ satisfies the classical null condition, it admits global smooth solutions for small data.