  The signum-Gordon model in 1+1 dimensions possesses the exact shockwave
solution with discontinuity of the field at the light cone and infinite
gradient energy. The energy of a regular part of the wave inside the light cone
is finite and it grows linearly with time. The initial data for such waves
contain a field configuration which is null in the space and has time
derivative proportional to the Dirac delta. We study regularized initial data
that lead to shock-like waves with finite gradient energy. We found that such
waves exist in the finite time intervals and finally they decay and produce a
cascade of oscillon-like structures. A pattern of the decay is very similar to
the one observed in process of scattering of compact oscillons.
