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 ${\partial }_{t}u\,=\,{\Delta}{u}^{m}+\vert x{\vert }^{\sigma }{u}^{p},$ posed in ${\mathbb{R}}^{N}$ with N ⩾ 3, where $0< m< {m}_{\mathrm{c}}=\frac{N-2}{N},p > 1,$ and the critical value for the weight $\sigma =\frac{2(p-1)}{1-m}.$ 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.