Propagation of Support and Singularity Formation for a Class of 2d Quasilinear Hyperbolic Systems

In this paper we consider a class of quasilinear, non-strictly hyper-bolic 2 2 systems in two space dimensions. Our main result is nite speed of propagation of the support of smooth solutions for these systems. As a consequence, we establish non-existence of global smooth solutions for a class of suuciently large, smooth initial data. The non-existence result applies to systems in conservation form, which satisfy a convexity condition on the uxes. We apply the non-existence result to a prototype example, obtaining an upper bound on the lifespan of smooth solutions with small amplitude initial data. We exhibit explicit smooth solutions for this example, obtaining the same upper bound on the lifespan and illustrating loss of smoothness through blow-up and through shock formation. Consider a quasilinear, hyperbolic system of diierential equations of the form: