Anomalous self-similar solutions of exponential type for the subcritical fast diffusion equation with weighted reaction

We prove existence and uniqueness of the branch of the so-called anomalous eternal solutions in exponential self-similar form for the subcritical fast-diffusion equation with a weighted reaction term ∂ t u = Δ u m + | x | σ u p, posed in R N with N ⩾ 3, where 1, > 0 1, and the critical value for the weight σ = 2 (p − 1) 1 − m. The branch of exponential self-similar solutions behaves similarly as the well-established anomalous solutions to the pure fast diffusion equation, but without a finite time extinction or a finite time blow-up, and presenting instead a change of sign of both self-similar exponents at m = m s = (N − 2) / (N + 2), leading to surprising qualitative differences. In this sense, the reaction term we consider realizes a perfect equilibrium in the competition between the fast diffusion and the reaction effects.