Weak solutions to a hyperbolic-elliptic problem

Abstract

We prove the existence of infinitely many local-in-time weak solutions to the initial-boundary value problem for a class of hyperbolic-elliptic equations in dimension $n\ge 2$ when the range of the magnitude of the initial spatial gradient overlaps with the unstable elliptic regime. Such solutions are extracted from the method of convex integration in a Baire category setup; they are smooth outside the phase mixing zone that is determined by a modified hyperbolic evolution, continuous on the space-time domain, and Lipschitz continuous in terms of the spatial variables.