Global self-similar solutions for Hardy-Hénon equations with linear and quasilinear diffusion

Abstract

Global self-similar solutions to the parabolic Hardy-Hénon equation $$ u_t=Δu^m+|x|^σu^p, \quad (x,t)\in\mathbb{R}^N\times(0,\infty), $$ are classified in the range of exponents $m\geq1$, $p>m$ and $σ>\max\{-2,-N\}$. The classification varies strongly with respect to the celebrated \emph{Fujita} and \emph{Sobolev critical exponents} $$ p_F(σ)=m+\frac{σ+2}{N}, \quad p_S(σ)= \begin{cases} \frac{m(N+2σ+2)}{N-2}, & \mbox{if } N\geq3, \\[1mm] \infty, & \mbox{if } N\in\{1,2\}. \end{cases} $$ Indeed, if $p\in(p_F(σ),p_S(σ))$, 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 $$ u(x,t)\sim C|x|^{-(σ+2)/(p-m)}, \quad {\rm as} \ |x|\to\infty. $$ If $p\geq p_S(σ)$, there are only self-similar solutions with the latter algebraic tail, while for $m<p\leq p_F(σ)$ 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-Hénon equations.

Figures (2)

• Figure 1: A shooting of various trajectories $l_C$ starting from $P_0$ and their corresponding profiles. Experiments for $m=2$, $N=3$, $p=5$ and $\sigma=1$.
• Figure 2: A shooting of various trajectories $l_C$ starting from $P_0$ and their corresponding profiles. Experiments for $m=2$, $N=3$, $p=15$ and $\sigma=1$.

Theorems & Definitions (18)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof : Proof of the statement related to $P_0$
• Lemma 2.2
• proof : Proof of the statement related to $Q_1$
• Lemma 2.3
• proof
• Lemma 2.4
• Lemma 2.5