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

TL;DR

The paper analyzes finite-time blow-up for the weighted parabolic equation $\partial_t u=\Delta u+|x|^{\sigma}u^p$ in $N\ge3$, $\sigma>-2$, $p>p_S(\sigma)$, and achieves a complete classification of backward self-similar blow-up profiles. By transforming the self-similar ODE into an autonomous 3D dynamical system and performing a detailed phase-space analysis, the authors construct profiles via backward shooting on center manifolds and across infinity, deriving critical thresholds such as $p_{JL}(\sigma)$ and Lepin-type exponents $p_L(\sigma)$ (and $\overline{p_L}(\sigma)$ for $\sigma<0$). They prove existence for all $p>p_S(\sigma)$ when $\sigma\ge2$, and multiplicity results: at least $k$ distinct self-similar blow-up solutions when $\sigma\ge4k-2$, highlighting a marked contrast with the homogeneous case ($\sigma=0$) where such solutions are absent for $p\ge p_L$. For $\sigma\in(0,2)$ and $\sigma\in(-2,0)$, the paper identifies Lépin-type windows $(p_S(\sigma),p_L(\sigma))$ and $(p_S(\sigma),\overline{p_L}(\sigma))$ in which self-similar blow-up solutions exist, supported by numerical evidence. The results illuminate how spatial weights modify blow-up structure, yield multiple self-similar regimes, and provide a robust dynamical-systems framework for reaction-diffusion with weighted sources.

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.

Figures (6)

• Figure 1: Trajectories in the invariant plane $\{X=0\}.$
• Figure 2: Trajectories in the invariant plane $\{Z=0\}.$
• Figure 3: Trajectories and profiles for $N=20$, $\sigma=1.5$, $p=10$, where $p_{JL}(\sigma)\approx3.55$ and $p_L(\sigma)=\infty$.
• Figure 4: Trajectories and profiles for $N=40$, $\sigma=1.5$, $p=10$, where $p_{JL}(\sigma)\approx1.39$ and $p_L(\sigma)=6.25$.
• Figure 5: Trajectories and profiles for $N=8$, $\sigma=-0.6$, $p=20$, where $p_{JL}(\sigma)\approx11.4$ and $\overline{p_L}(\sigma)=\infty$.

Theorems & Definitions (32)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4