SINGULAR SOLUTIONS OF PARABOLIC p-LAPLACIAN WITH ABSORPTION

We consider, for p ∈ (1, 2) and q > 1, the p-Laplacian evolution equation with absorption ut = div (|∇u|p−2∇u)− u in R × (0,∞). We are interested in those solutions, which we call singular solutions, that are non-negative, non-trivial, continuous in Rn × [0,∞) \ {(0, 0)}, and satisfy u(x, 0) = 0 for all x = 0. We prove the following: (i) When q ≥ p− 1 + p/n, there does not exist any such singular solution. (ii) When q < p − 1 + p/n, there exists, for every c > 0, a unique singular solution u = uc that satisfies ∫ Rn u(·, t) → c as t ↘ 0. Also, uc ↗ u∞ as c ↗ ∞, where u∞ is a singular solution that satisfies ∫ Rn u∞(·, t) → ∞ as t ↘ 0. Furthermore, any singular solution is either u∞ or uc for some finite positive c.