  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.
Therefore, we will work with large data (in energy norm). Moreover, no symmetry
condition is imposed on the initial datum. We emphasize the geometric
perspectives of shock formations in the proof. More specifically, the key idea
is to study the interplay between the following two objects: (1) the energy
estimates of the linearized equations of $(\star)$; (2) the differential
geometry of the Lorentzian metric $g=-\dfrac{1}{\left(1+3G^{\prime\prime}(0)
(\partial_t\phi)^2\right)} d t^2+dx_1^2+dx_2^2+dx_3^2$. Indeed, the study of
the characteristic hypersurfaces (implies shock formation) is the study of the
null hypersurfaces of $g$. The techniques in the proof are inspired by the work
\cite{Ch-Shocks} in which the formation of shocks for $3$-dimensional
relativistic compressible Euler equations with small initial data is
established. We also use the short pulse method which is introduced in the
study of formation of black holes in general relativity in \cite{Ch-BlackHoles}
and generalized in \cite{K-R-09}.
