Singular Solutions to the Heat Equations With Nonlinear Absorption and Hardy Potentials

We study the existence and nonexistence of singular solutions to the equation [Formula: see text], p > 1, in ℝN× [0, ∞), N ≥ 3, with a singularity at the point (0, 0), that is, nonnegative solutions satisfying u(x, 0) = 0 for x ≠ 0, assuming that α > -2 and [Formula: see text]. The problem is transferred to the one for a weighted Laplace–Beltrami operator with a nonlinear absorption, absorbing the Hardy potential in the weight. A classification of a singular solution to the weighted problem either as a source solution with a multiple of the Dirac mass as initial datum, or as a unique very singular solution, leads to a complete classification of singular solutions to the original problem, which exist if and only if [Formula: see text].