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

Abstract

Global self-similar solutions to the parabolic Hardy-Hénon equation $$u_t=\Delta u^m+|x{|}^{\sigma }{u}^p,(x,t)\in {\mathbb{R}}^N\times (0,\infty ),$$ 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} $$ p_F(σ)=m+\frac{σ+2}{N}, \quad p_S(σ)= \begin{cases} \frac{m(N+2σ+2)}{N-2}, & \mbox{if } N\geq3, \\[1mm] \infty, & \mbox{if } N\in\{1,2\}. \end{cases} $$ 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 $|x|\to \infty$ (if $m=1$), as well as a one-parameter family satisfying $$u(x,t)\sim C|x{|}^{-(\sigma +2)/(p-m)},\mathrm{as}|x|\to \infty .$$ 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.