A new transformation for the subcritical fast diffusion equation with source and applications

A new transformation for radially symmetric solutions to the subcritical fast diffusion equation with spatially inhomogeneous source $${\mathrm{\partial }}_{t}u=\mathrm{\Delta }{u}^m+\mathrm{\mid }x{\mathrm{\mid }}^{\sigma }{u}^p,$$ posed for $(x,t)\in {\mathbb{R}}^N\times (0,\mathrm{\infty })$ and with dimension and exponents $$N\ge 3,0<m<m_c:=\frac{N-2}{N},\sigma \in (-2,\mathrm{\infty }),$$ is introduced. It plays a role of a kind of symmetry with respect to the critical exponents $$m_s=\frac{N-2}{N+2},p_L(\sigma )=1+\frac{\sigma (1-m)}{2},p_s(\sigma )=\frac{m(N+2\sigma +2)}{N-2}\mathrm{.}$$ This transformation is then applied for classifying self-similar solutions with or without finite time blow-up to the subcritical fast diffusion equation with source when $p>\max \{1,p_L(\sigma )\}$, having as starting point previous results by the authors.