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Global solutions versus finite time blow-up for the supercritical fast diffusion equation with inhomogeneous source

Razvan Gabriel Iagar, Ariel Sánchez

TL;DR

This work classifies self-similar solutions to the supercritical fast diffusion equation with a spatially inhomogeneous source, $\\partial_tu=\\Delta u^m+|x|^\\sigma u^p$, by balancing diffusion and source terms through phase-space analysis. The authors identify sharp distinctions governed by Fujita-type and Sobolev-type exponents, $p_F(\\sigma)$ and $p_s(\\sigma)$, determining when global forward self-similar solutions exist (for $p\\in(p_F(\\sigma),p_s(\\sigma))$) and when finite-time blow-up self-similar profiles arise (for $\\sigma\\in(-2,0)$ and $p\\in(\\max\\{1,p_L(\\sigma)\\},p_s(\\sigma))$). They prove nonexistence of backward blow-up profiles for $\\sigma\\geq0$ within the same $p$-range and establish a constructive existence result for $\\sigma\\in(-2,0)$ via detailed phase-plane analysis, including explicit tail and singular behaviors. The results are new even in the homogeneous case $\\sigma=0$ and are extended to $N=1,2$ with appropriate adaptations, providing a comprehensive dynamical-systems framework for these nonlinear diffusion with inhomogeneous sources.

Abstract

Solutions in self-similar form, either global in time or presenting finite time blow-up, to the supercritical fast diffusion equation with spatially inhomogeneous source $$ \partial_tu=Δu^m+|x|^σu^p, \quad (x,t)\in\mathbb{R}^N\times(0,\infty) $$ with $$ m_c=\frac{(N-2)_+}{N}\leq m<1, \quad σ\in(\max\{-2,-N\},\infty), \quad p>\max\left\{1+\frac{σ(1-m)}{2},1\right\} $$ are considered. It is proved that global self-similar solutions with the specific tail behavior $$ u(x,t)\sim C(m)|x|^{-2/(1-m)}, \qquad {\rm as} \ |x|\to\infty $$ exist exactly for $p\in(p_F(σ),p_s(σ))$, where $$ p_F(σ)=m+\frac{σ+2}{N}, \qquad p_s(σ)=\left\{\begin{array}{ll}\frac{m(N+2σ+2)}{N-2}, & N\geq3,\\\infty, & N\in\{1,2\}, \end{array}\right. $$ are the renowned Fujita and Sobolev critical exponents. In contrast, it is shown that self-similar solutions presenting finite time blow-up exist for any $σ\in(-2,0)$ and $p$ as above, but do not exist for any $σ\geq0$ and $p\in(p_F(σ),p_s(σ))$. We stress that all these results are \emph{new also in the homogeneous case $σ=0$}.

Global solutions versus finite time blow-up for the supercritical fast diffusion equation with inhomogeneous source

TL;DR

This work classifies self-similar solutions to the supercritical fast diffusion equation with a spatially inhomogeneous source, , by balancing diffusion and source terms through phase-space analysis. The authors identify sharp distinctions governed by Fujita-type and Sobolev-type exponents, and , determining when global forward self-similar solutions exist (for ) and when finite-time blow-up self-similar profiles arise (for and ). They prove nonexistence of backward blow-up profiles for within the same -range and establish a constructive existence result for via detailed phase-plane analysis, including explicit tail and singular behaviors. The results are new even in the homogeneous case and are extended to with appropriate adaptations, providing a comprehensive dynamical-systems framework for these nonlinear diffusion with inhomogeneous sources.

Abstract

Solutions in self-similar form, either global in time or presenting finite time blow-up, to the supercritical fast diffusion equation with spatially inhomogeneous source with are considered. It is proved that global self-similar solutions with the specific tail behavior exist exactly for , where are the renowned Fujita and Sobolev critical exponents. In contrast, it is shown that self-similar solutions presenting finite time blow-up exist for any and as above, but do not exist for any and . We stress that all these results are \emph{new also in the homogeneous case }.
Paper Structure (7 sections, 16 theorems, 158 equations, 2 figures)

This paper contains 7 sections, 16 theorems, 158 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The supercritical range. Phase plane and local analysis
  3. Proof of Theorem \ref{['th.global.super']}
  4. Blow-up profiles. General facts
  5. Existence of blow-up profiles for $\sigma\in(-2,0)$
  6. Non-existence of blow-up profiles for $m\in[m_c,1)$ and $\sigma\geq0$
  7. Dimensions $N=1$ and $N=2$

Key Result

Theorem 1.1

Let $m\in[m_c,1)$ and $\sigma\in(\max\{-2,-N\},\infty)$. Then

Figures (2)

  • Figure 1: Phase portrait with orbits on the unstable manifold of $P_0$. Experiment for $m=2/3$, $p=3$, $\sigma=1$, $N=3$.
  • Figure 2: The plane defined in \ref{['plane1']} and the surface defined in \ref{['surface']}.

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 20 more