A quasilinear wave with a supersonic shock in a weak solution interrupting the classical development
We study the Cauchy problem for classical and weak shock-forming solutions to a model quasilinear wave equation in $1+1$ dimensions arising from a convenient choice of $C^{\infty}$ initial data, which allows us to solve the equation using elementary arguments. The simplicity of our model allows us to succinctly illustrate various phenomena of geometric and analytic significance tied to shocks, which we view as a prototype for phenomena that can occur in more general quasilinear hyperbolic PDE solutions. Our Cauchy problem admits a classical solution that blows up in finite time. The classical solution is defined in a largest possible globally hyperbolic region called a maximal globally hyperbolic development (MGHD), and its properties are tied to the intrinsic Lorentzian geometry of the equation and solution. The boundary of the MGHD contains an initial singularity, a singular boundary along which the solution's second derivatives blow up (the solution and its first derivatives remain bounded), and a Cauchy horizon. Our main results provide the first example of a provably unique MGHD for a shock-forming quasilinear wave equation solution; it is provably unique because its boundary has a favorable global structure that we precisely describe. We also prove that for the same $C^{\infty}$ initial data, the Cauchy problem admits a second kind of solution: a unique global weak entropy solution that has a shock curve separating two smooth regions. Of particular interest is our proof that the classical and weak solutions agree before the shock but differ in a region to the future of the first singularity where both solutions are defined.