ABSTRACT
We consider quasilinear PDEs which are second-order in the time variable, i.e., contain the derivative utt . This class includes the well-known second and higher-order PDEs of hyperbolic type, which provide us with interesting new examples of formation of evolution singularities. There are many applications of quasilinear hyperbolic equations possessing exact and explicit solutions. We present a few of equations below and include more references in the Remarks. In particular, Zabusky [589] proposed to use the hyperbolic equation
utt = k(ux)ux x as a model for the dynamics of nonlinear strings. In this context, it is worth mentioning the standard derivation of the vibrating string equation by Newton’s Second Law for a homogeneous thin string. Under the assumption that Hook’s Law applies, this yields, in the dimensionless form, the quasilinear PDE
utt = [
] x .
