On the formation of shock for quasilinear wave equations by pulse with weak intensity

Abstract

In this paper we continue to study the shock formation for the $3$-dimensional quasilinear wave equation \begin{align}\label{main eq} -(1+3G"(0)(\partial_{t}\phi)^{2})\partial^{2}_{t}\phi+\Delta\phi=0,\tag{\textbf{$\star$}} \end{align} with $G"(0)$ being a non-zero constant. Since \eqref{main eq} admits global-in-time solution with small initial data, to present shock formation, we consider a class of large data. Moreover, no symmetric assumption is imposed on the data. Compared to our previous work [18], here we pose data on the hypersurface $\{(t,x)|t=-r_{0}\}$ instead of $\{(t,x)|t=-2\}$, with $r_{0}$ being arbitrarily large. We prove an a priori energy estimate independent of $r_{0}$. Therefore a complete description of the solution behavior as $r_{0}\rightarrow\infty$ is obtained. This allows us to relax the restriction on the profile of initial data which still guarantees shock formation. Since \eqref{main eq} can be viewed as a model equation for describing the propagation of electromagnetic waves in nonlinear dielectric, the result in this paper reveals the possibility to use wave pulse with weak intensity to form electromagnetic shocks in laboratory. A main new feature in the proof is that all estimates in the present paper do \emph{not} depend on the parameter $r_{0}$, which requires different methods to obtain energy estimates. As a byproduct, we prove the existence of semi-global-in-time solutions which lead to shock formation by showing that the limits of the initial energies exist as $r_{0}\rightarrow\infty$. The proof combines the ideas in [5] where the the formation of shocks for 3-dimensional relativistic compressible Euler equations with small initial data is established, and the short pulse method introduced in [6] and generalized in [15], where the formation of black holes in general relativity is proved.