Global self-similar solutions for Hardy-H\'enon equations with linear and quasilinear diffusion

Abstract

Global self-similar solutions to the parabolic Hardy-H\'enon equation $$ u_t=\Delta u^m+|x|^{\sigma}u^p, \quad (x,t)\in\mathbb{R}^N\times(0,\infty), $$ are classified in the range of exponents $m\geq1$, $p>m$ and $\sigma>\max\{-2,-N\}$. The classification varies strongly with respect to the celebrated \emph{Fujita} and \emph{Sobolev critical exponents} $$ p_F(\sigma)=m+\frac{\sigma+2}{N}, \quad p_S(\sigma)= \begin{cases} \frac{m(N+2\sigma+2)}{N-2}, & \mbox{if } N\geq3, \\[1mm] \infty, & \mbox{if } N\in\{1,2\}. \end{cases} $$ 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 $$ u(x,t)\sim C|x|^{-(\sigma+2)/(p-m)}, \quad {\rm as} \ |x|\to\infty. $$ If $p\geq p_S(\sigma)$, there are only self-similar solutions with the latter algebraic tail, while for $m<p\leq p_F(\sigma)$ 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-H\'enon equations.