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

Abstract

We prove existence and uniqueness of the branch of the so-called \emph{anomalous eternal solutions} in exponential self-similar form for the subcritical fast-diffusion equation with a weighted reaction term $$ \partial_tu=\Delta u^m+|x|^{\sigma}u^p, $$ posed in $\real^N$ with $N\geq3$, where $$ 0<m<m_c=\frac{N-2}{N}, \qquad p>1, $$ 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 \emph{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 \emph{perfect equilibrium} in the competition between the fast diffusion and the reaction effects.