ABSTRACT
In this chapter, we study formation of new basic blow-up singularities in higher-order quasilinear hyperbolic (wave-type) equations. As a basic model, we consider the fourth-order Boussinesq-type, or a quasilinear wave equation (the QWE-4, in short) of the form
utt = −(|u|nu)xxxx in IR× (0, T ), with an exponent n > 0. (1) In particular, n = 2 yields the cubic equation with an analytic nonlinearity:
utt = −(u3)xxxx. (2) Of course, as is well-known in PDE theory and as we have already seen several times, having analytic nonlinearities does not imply any smooth properties
and
of the solutions, since this PDE is degenerate at the zero level {u = 0}. Moreover, since it is hyperbolic, this implies certain dispersion phenomena, and, as a result, leads to a formation (for the first time) of shocks, which is the main new issue of this chapter.
