[Paper Review] Anomalous self-similar solutions of exponential type for the subcritical fast diffusion equation with weighted reaction

Ariel Sánchez, Razvan Gabriel Iagar|arXiv (Cornell University)|Jun 16, 2022

Mathematical and Theoretical Epidemiology and Ecology Models34 references10 citations

TL;DR

This paper establishes the existence and uniqueness of anomalous eternal self-similar solutions of exponential type for the subcritical fast diffusion equation with weighted reaction $\partial_t u = \Delta u^m + |x|^\sigma u^p$ in $\mathbb{R}^N$, $N \geq 3$, where $0 < m < m_c = (N-2)/N$, $p > 1$, and $\sigma = 2(p-1)/(1-m)$. The key result is that these solutions neither extinguish nor blow up in finite time, achieving a perfect balance between diffusion and reaction, with a sign change in self-similar exponents at $m = m_s = (N-2)/(N+2)$, leading to qualitative differences from classical anomalous solutions.

ABSTRACT

We prove existence and uniqueness of the branch of the so-called anomalous eternal solutions in exponential self-similar form for the subcritical fast-diffusion equation with a weighted reaction term ∂ t u = Δ u m + | x | σ u p , posed in R N with N ⩾ 3, where 1,$&gt; 0 &lt; m &lt; m c = N − 2 N , p &gt; 1 , and the critical value for the weight σ = 2 ( p − 1 ) 1 − m . The branch of exponential self-similar solutions behaves similarly as the well-established anomalous solutions to the pure fast diffusion equation, but without a finite time extinction or a finite time blow-up, and presenting instead a change of sign of both self-similar exponents at m = m s = ( N − 2)/( N + 2), leading to surprising qualitative differences. In this sense, the reaction term we consider realizes a perfect equilibrium in the competition between the fast diffusion and the reaction effects.

Motivation & Objective

• To establish the existence and uniqueness of eternal self-similar solutions in exponential form for the subcritical fast diffusion equation with weighted reaction term.
• To analyze the dynamics of the equation when the reaction weight $\sigma$ is tuned to the critical value $\sigma = 2(p-1)/(1-m)$, which balances diffusion and reaction effects.
• To show that these solutions do not exhibit finite-time extinction or blow-up, unlike classical anomalous solutions.
• To investigate the qualitative behavior of the self-similar exponents and their dependence on the parameter $m$, particularly at the critical value $m_s = (N-2)/(N+2)$.

Proposed method

• Derive the self-similar transformation $u(x,t) = e^{\alpha t} f(e^{\beta t} x)$ to reduce the PDE to an ODE system in the phase plane.
• Perform phase plane analysis on the resulting autonomous ODE system to study the critical points and orbits corresponding to self-similar profiles.
• Use dynamical systems techniques, including center manifold analysis and saddle-saddle connections, to prove existence and uniqueness of the anomalous solution branch.
• Construct explicit solutions for special cases, such as $m = m_s$, to verify the general results and illustrate the self-map symmetry.
• Establish a self-map between solutions in the range $m \in (0, m_s)$ and $m \in (m_s, m_c)$, showing symmetry under dimension and parameter transformation.
• Analyze the behavior of solutions at infinity and at the origin, including vertical asymptotes and tail decay rates, to classify solution profiles.

Experimental results

Research questions

• RQ1 Does a branch of anomalous self-similar solutions exist for the subcritical fast diffusion equation with weighted reaction when $\sigma = 2(p-1)/(1-m)$?
• RQ2 Do these solutions avoid both finite-time extinction and finite-time blow-up, achieving a perfect balance between diffusion and reaction?
• RQ3 How does the behavior of the self-similar exponents $\alpha(m)$ and $\beta(m)$ change at $m = m_s = (N-2)/(N+2)$, and what qualitative differences does this produce?
• RQ4 Can explicit solutions be constructed that illustrate the phase plane structure and the role of critical points such as $P_0$, $P_1$, $P_2$, and $Q_4$?
• RQ5 Is there a self-map that relates solutions in the range $m \in (0, m_s)$ to those in $m \in (m_s, m_c)$, and how does it preserve the solution structure?

Key findings

• The paper proves the existence and uniqueness of a branch of anomalous self-similar solutions in exponential form for $0 < m < m_c$, with the critical weight $\sigma = 2(p-1)/(1-m)$, which prevents both finite-time extinction and blow-up.
• The self-similar exponents $\alpha(m)$ and $\beta(m)$ change sign at $m = m_s = (N-2)/(N+2)$, leading to a qualitative change in solution behavior, distinguishing them from classical anomalous solutions.
• For $m = m_s$, the solution is explicit with $\beta(m_s) = 0$ and $\alpha(m_s) = (N+2)/4$, corresponding to a stationary profile.
• Explicit solutions are constructed for special cases: one with a vertical asymptote at the origin ($f(\xi) = C\xi^{-2/(1-m)}$), and another family with two vertical asymptotes depending on the sign of a free parameter $D$.
• A self-map is constructed that maps solutions in $m \in (0, m_s)$ to those in $m \in (m_s, m_c)$, preserving the phase plane structure and symmetry, generalizing known self-maps for the standard fast diffusion equation.
• The phase plane analysis reveals saddle-saddle connections $P_0$–$P_1$ that correspond to the anomalous solution branch, and other orbits connecting $P_2$ to $Q_4$ or $P_1$, corresponding to different solution profiles with vertical asymptotes.