Self-similar blow-up solutions for the supercritical parabolic Hardy-H\'enon equation

Abstract

We classify the self-similar solutions presenting finite time blow-up to the parabolic Hardy-H\'enon equation $$ \partial_tu=\Delta u+|x|^{\sigma}u^p, \quad (x,t)\in\mathbb{R}^N\times(0,\infty), $$ in dimension $N\geq3$ and the range of exponents $$ \sigma\in(-2,\infty), \quad p>p_S(\sigma):=\frac{N+2\sigma+2}{N-2}. $$ We establish the \emph{existence of self-similar blow-up solutions for any $p>p_S(\sigma)$}, provided $\sigma\geq2$. Moreover, we prove that, if $k$ is any natural number and $\sigma\geq 4k-2$, the parabolic Hardy-H\'enon equation has at least $k$ different self-similar blow-up solutions for any $p>p_S(\sigma)$. These results are in a stark contrast with the standard reaction-diffusion equation $$ \partial_tu=\Delta u+u^p, \quad (x,t)\in\mathbb{R}^N\times(0,\infty), $$ 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 $\sigma\in(-2,2)$, we derive the expression of generalized Lepin exponents $p_L(\sigma)$ for $\sigma\in(0,2)$, respectively $\overline{p_L}(\sigma)$ for $\sigma\in(-2,0)$, and prove existence of self-similar solutions with finite time blow-up for $p\in(p_S(\sigma),p_L(\sigma))$, respectively $p\in(p_S(\sigma),\overline{p_L}(\sigma))$. Numerical evidence of the optimality of these exponents is also included.