S. Kichenassamy et W. Littman, BLOW-UP SURFACES FOR NONLINEAR-WAVE EQUATIONS .2., Communications in partial differential equations, 18(11), 1993, pp. 1869-1899
In this second part, we prove that the equation square u = e(u) has so
lutions blowing up near a point of any analytic, space-like hypersurfa
ce in R(n), without any additional condition; if (phi(x,t) = 0) is the
equation of the surface, u - ln(2/phi2) is not necessarily analytic,
and generally contains logarithmic terms. We then construct singular s
olutions of general semilinear equations which blow-up on a non-charac
teristic surface, provided that the first term of an expansion of such
solutions can be found. We finally list a few other simple nonlinear
evolution equations to which our methods apply; in particular, formal
solutions of soliton equations given by a number of authors can be sho
wn to be convergent by this procedure.