Self-similar blow-up solutions for the supercritical parabolic Hardy-Hénon equation

Abstract

We classify the self-similar solutions presenting finite time blow-up to the parabolic Hardy-Hénon equation $$ \partial_tu=Δu+|x|^σu^p, \quad (x,t)\in\mathbb{R}^N\times(0,\infty), $$ in dimension $N\geq3$ and the range of exponents $$ σ\in(-2,\infty), \quad p>p_S(σ):=\frac{N+2σ+2}{N-2}. $$ We establish the \emph{existence of self-similar blow-up solutions for any $p>p_S(σ)$}, provided $σ\geq2$. Moreover, we prove that, if $k$ is any natural number and $σ\geq 4k-2$, the parabolic Hardy-Hénon equation has at least $k$ different self-similar blow-up solutions for any $p>p_S(σ)$. These results are in a stark contrast with the standard reaction-diffusion equation $$ \partial_tu=Δu+u^p, \quad (x,t)\in\mathbb{R}^N\times(0,\infty), $$ for which non-existence of any self-similar solution has been established, provided $p$ overpasses the Lepin exponent $p_L:=1+\frac{6}{N-10}$, $N\geq11$. For $σ\in(-2,2)$, we derive the expression of generalized Lepin exponents $p_L(σ)$ for $σ\in(0,2)$, respectively $\overline{p_L}(σ)$ for $σ\in(-2,0)$, and prove existence of self-similar solutions with finite time blow-up for $p\in(p_S(σ),p_L(σ))$, respectively $p\in(p_S(σ),\overline{p_L}(σ))$. Numerical evidence of the optimality of these exponents is also included.