Global self-similar solutions for Hardy-Hénon equations with linear and quasilinear diffusion

Global self-similar solutions to the parabolic Hardy-Hénon equation $$u_t=\mathrm{\Delta }{u}^m+\mathrm{\mid }x{\mathrm{\mid }}^{\sigma }{u}^p,(x,t)\in {\mathbb{R}}^N\times (0,\mathrm{\infty }),$$ are classified in the range of exponents $m\ge 1$, $p>m$ and $\sigma >\max \{-2,-N\}$. The classification varies strongly with respect to the celebrated \emph{Fujita} and \emph{Sobolev critical exponents} $$p_F(\sigma )=m+\frac{\sigma +2}{N},p_S(\sigma )=\{\begin{array}{ll}\frac{m(N+2\sigma +2)}{N-2},& ifN\ge 3,\\ \mathrm{\infty },& ifN\in \{1,2\}\mathrm{.}\end{array}$$ Indeed, if $p\in (p_F(\sigma ),p_S(\sigma ))$, both equations admit self-similar solutions with either compact support (if $m>1$) or Gaussian-like tail as $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$ (if $m=1$), as well as a one-parameter family satisfying $$u(x,t)\sim C\mathrm{\mid }x{\mathrm{\mid }}^{-(\sigma +2)\mathrm{/}(p-m)},\mathrm{a}\mathrm{s}\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }\mathrm{.}$$ If $p\ge p_S(\sigma )$, there are only self-similar solutions with the latter algebraic tail, while for $m<p\le p_F(\sigma )$ no global solutions exist. The results open the way for a deeper study of the role of these solutions in the dynamics of the Hardy-Hénon equations.