This paper deals with the extending of the existing theory of nonnegative solutions of the nonlinear evolution equation \[ u_t=\Delta (u^m),\;0< m<1, \] which serves as a mathematical model for the interplay of fast and slow propagation speeds in an evolution process of diffusive type. The goal of the authors is to formulate a theory of existence, uniqueness and continuous dependence of solutions with arbitrarily large data in a suitable class of large data in a suitable class of large solutions, as well as the inverse problem of assigning an initial trace to any given solution. The authors show that for the range \[ m_c<m<1,\;m_c= \max\left\{ {N-2\over N},0 \right\} \] three closely related problems can be solved in an optimal way. The authors emphasize the main novelty of their study, namely the existence of strong singularities which behave like permanent sources of radiation.