Formation of Singularities for Hyperbolic Differential Equations With Geometric Backgrounds

ABSTRACT In this paper, we consider the formation of singularities of the Cauchy problem for hyperbolic differential equations arising from a number of geometric problems, which includes the hyperbolic mean curvature flow and so on. The initial data have been assumed to be periodic, and thus, the smooth solutions can persist only in a short time. Under weaker conditions on the initial data, we are able to prove that the singularities of solutions will form inevitably. Moreover, the estimate on the lifespan is provided in detail.