In this paper we discuss the existence and uniqueness of singular solutions for equations of the form $$ (F) u_t = u{xx} - $mid$u$mid$^{q-1} u_x - $mid$u$mid$^{p-1}u, p,q > 1, $$</TEX> in the domain $Q = {(x,t) : x in R, t > 0}$. This equation represents a model of diffusion-convection with absorption.