Let Omega be a bounded domain of R-N (N >= 2). We obtain a necessary and a sufficient condition, expressed in terms of capacities, for the existence of a solution to the porous medium equation with absorption {u(t) - Delta (vertical bar u vertical bar(m-1)u) + vertical bar u vertical bar(q-1)u = mu in Omega x (0, T) u = 0 on partial derivative Omega x (0, T) u(0) = sigma where sigma and mu are bounded Radon measures, q > max(m, 1), and m > N-2/N. Wealso obtain a sufficient condition for the existence of a solution to the p-Laplace evolution equation {u(t) - Delta(p)u + vertical bar u vertical bar(q-1)u = mu in Omega x (0, T) u = 0 on partial derivative Omega x (0, T) u(0) = sigma where q > p - 1 and p > 2.