It's well known that solutions of genuinely nonlinear hyperbolic PDEs lead to shock singularities in finite time, under very weak assumptions on the initial data. However, proofs of this statement invariably assume uniformity of the PDE coefficients in space and time. What if the coefficients are allowed to vary, as would be the case for waves in many real materials, whose properties may be random or periodic?
Surprisingly little is known about the answer to this question, but a first attempt to answer it in part is made in my recently submitted manuscript "Shock dynamics in layered periodic media"
. Among the "shocking" findings:
-For certain media and relatively general initial conditions, shock formation seems not to occur even after extremely long times.
-Shocks that would be stable in a homogeneous medium are frequently not stable in a heterogeneous medium
-The asymptotic behavior of solutions in heterogeneous media is generally different; rather than consisting of N-waves, the solutions may be composed of solitary waves, for instance.
To get an idea of what's going on, take a look at some movies
showing animations of the remarkable behavior of the solution.