[論文レビュー] Global self-similar solutions for Hardy-Hénon equations with linear and quasilinear diffusion

Razvan Gabriel Iagar, Ariel G. Sánchez+1 more|arXiv (Cornell University)|Feb 24, 2026

Nonlinear Partial Differential Equations被引用数 0

ひとこと要約

The paper classifies global self-similar solutions to Hardy-Hénon parabolic equations with linear and quasilinear diffusion, detailing how existence and tail behavior depend on p, m, and σ relative to Fujita and Sobolev exponents.

ABSTRACT

Global self-similar solutions to the parabolic Hardy-Hénon equation $$ u_t=Δu^m+|x|^σu^p, \quad (x,t)\in\mathbb{R}^N imes(0,\infty), $$ 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 $|x| o\infty$ (if $m=1$), as well as a one-parameter family satisfying $$ u(x,t)\sim C|x|^{-(σ+2)/(p-m)}, \quad { m as} \ |x| o\infty. $$ If $p\geq p_S(σ)$, there are only self-similar solutions with the latter algebraic tail, while for $m

研究の動機と目的

• Classify global in time radially symmetric self-similar solutions for the parabolic Hardy-Hénon equations in the unified form with m ≥ 1, p > pF(σ), and σ > max{-2, -N}.
• Determine how solution profiles behave at infinity (Gaussian-like vs algebraic tails) and when compact support occurs, depending on p relative to pF(σ) and pS(σ).
• Relate the profile behavior to the diffusion type (linear m=1 vs quasilinear m>1) and establish existence/non-existence regimes for global self-similar solutions.

提案手法

• Seek self-similar solutions u(x,t)=t^{-α}f(|x|t^{-β}) and derive the profile equation (f^m)''+(N-1)/ξ (f^m)' + α f + β ξ f' + ξ^σ f^p = 0.
• Compute α and β from balancing time derivatives: α=(σ+2)/(σ(m-1)+2(p-1)) and β=(p-m)/(σ(m-1)+2(p-1)).
• Reformulate the profile equation as a 3D dynamical system in variables X,Y,Z with a logarithmic time change to analyze near 0 and ∞.
• Perform a phase-space (finite and infinite) analysis including critical points P0, P1, P2 and points at infinity Q1,...,Qγ on the Poincaré hypersphere.
• classify trajectories to identify profile behaviors: Gaussian-like decay, algebraic tails, and compact support depending on p relative to pF(σ) and pS(σ).
• Utilize center manifold theory and invariant-plane arguments to understand local and asymptotic behavior of profiles.

実験結果

リサーチクエスチョン

• RQ1 What global self-similar profiles exist for the parabolic Hardy-Hénon equations with m≥1 and p>pF(σ)?
• RQ2 How do tail behaviors of self-similar profiles depend on p relative to pF(σ) and pS(σ) and on whether diffusion is linear (m=1) or quasilinear (m>1)?
• RQ3 Do compactly supported or Gaussian-like tails occur, and under what parameter ranges?
• RQ4 What nonexistence or existence results can be established for global self-similar solutions in different regimes?

主な発見

• For p in (pF(σ), pS(σ)) and m=1, there exist A*<A* (with A* finite) such that f(ξ;A*) decays Gaussian-like as ξ→∞, while f(ξ;A) with A<A* decays algebraically.
• For p≥pS(σ), all profiles have algebraic tails, while in the m=1 case there is a unique Gaussian-like profile in the intermediate range.
• For m>1 and p in (pF(σ), pS(σ)), there exists a profile f(·;A*) that is compactly supported, and for A<A* profiles have algebraic tails with f(ξ) ~ L(A) ξ^{-(σ+2)/(p-m)} as ξ→∞.
• For N≥3 and p≥pS(σ), all profiles exhibit the algebraic tail behavior described above.
• In the full range p>pF(σ), if p≥pS(σ) then Gaussian-like tails disappear and algebraic tails dominate; no global solutions exist in the regime 1<p≤pF(σ).
• The analysis provides a dynamical-systems framework (finite and infinite critical points) to categorize the possible self-similar profiles and their asymptotics.