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 $${\partial }_{t}u=\Delta u+|x{|}^{\sigma }{u}^p,(x,t)\in {\mathbb{R}}^N\times (0,\infty ),$$ in dimension $N\ge 3$ and the range of exponents $$\sigma \in (-2,\infty ),p>p_S(\sigma ):=\frac{N+2\sigma +2}{N-2}.$$ We establish the \emph{existence of self-similar blow-up solutions for any $p>p_S(\sigma )$}, provided $\sigma \ge 2$. Moreover, we prove that, if $k$ is any natural number and $\sigma \ge 4k-2$, the parabolic Hardy-Hénon 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 $${\partial }_{t}u=\Delta u+u^p,(x,t)\in {\mathbb{R}}^N\times (0,\infty ),$$ for which non-existence of any self-similar solution has been established, provided $p$ overpasses the Lepin exponent $p_L:=1+\frac{6}{N-10}$, $N\ge 11$. 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.