[Paper Review] Extinction behaviour for the fast diffusion equations with critical exponent and Dirichlet boundary conditions

Yannick Sire, Juncheng Wei+1 more|arXiv (Cornell University)|Jun 1, 2020

Nonlinear Partial Differential Equations24 references4 citations

TL;DR

This paper establishes the rigorous extinction behavior for the fast diffusion equation with critical exponent $ m = \frac{n-2}{n+2} $ and Dirichlet boundary conditions on general smooth bounded domains in $ \mathbb{R}^n $, $ n \geq 3 $. Using a parabolic gluing method, it constructs solutions that vanish at finite time $ T $ with a precise asymptotic profile involving logarithmic corrections, confirming and extending a conjecture by Galaktionov and King for the radially symmetric case.

ABSTRACT

For a smooth bounded domain $Ω\subseteq\mathbb{R}^n$, $n\geq 3$, we consider the fast diffusion equation with critical sobolev exponent $$\frac{\partial w}{\partialτ} =Δw^{\frac{n-2}{n+2}}$$ under Dirichlet boundary condition $w(\cdot, τ) = 0$ on $\partialΩ$. Using the parabolic gluing method, we prove existence of an initial data $w_0$ such that the corresponding solution has extinction rate of the form $$\|w(\cdot, τ)\|_{L^\infty(Ω)} = γ_0(T-τ)^{\frac{n+2}{4}}\left|\ln(T-τ) ight|^{\frac{n+2}{2(n-2)}}(1+o(1))$$ as $t o T^-$, here $T > 0$ is the finite extinction time of $w(x, τ)$. This generalizes and provides rigorous proof of a result of Galaktionov and King \cite{galaktionov2001fast} for the radially symmetric case $Ω=B_1(0) : = \{x\in \mathbb{R}^n||x| < 1\}\subset\mathbb{R}^n$.

Motivation & Objective

• To provide a rigorous asymptotic analysis of the extinction behavior for the fast diffusion equation with critical exponent $ m = \frac{n-2}{n+2} $ in general smooth bounded domains.
• To extend the formal results of Galaktionov and King, which were limited to the radially symmetric case, to arbitrary smooth domains.
• To establish the existence of initial data leading to a specific extinction rate with logarithmic corrections.
• To characterize the blow-up profile of the solution near extinction time using a multi-peak structure centered at fixed points in the domain.

Proposed method

• Employing the parabolic gluing method to construct a solution with prescribed asymptotic behavior near extinction time.
• Decomposing the solution into inner and outer regions, with the inner region capturing the localized peak behavior near $ k $ fixed points $ q_j \in \Omega $.
• Using a matched asymptotic expansion approach, where the inner solution is modeled on the standard ground state $ U(y) $, and the outer solution is governed by the Green's function with Dirichlet boundary conditions.
• Introducing parameter functions $ \tilde{\mu}_j(\tau) $ and $ \tilde{\xi}_j(\tau) $ to describe the amplitude and location of the peaks, with $ \tilde{\mu}_j \sim \beta_j \left(\log \frac{T}{T-\tau}\right)^{-1/(n-2)} $.
• Solving the inner and outer problems via a fixed-point argument in a weighted function space, ensuring orthogonality and decay conditions.
• Establishing solvability of the resulting ODE system for the parameters through contraction mapping and perturbation theory.

Experimental results

Research questions

• RQ1 What is the precise extinction rate of solutions to the fast diffusion equation with critical exponent $ m = \frac{n-2}{n+2} $ in general smooth bounded domains?
• RQ2 How does the solution profile behave near the extinction time $ T $, particularly in terms of peak localization and amplitude decay?
• RQ3 Can the formal asymptotic results of Galaktionov and King for the radially symmetric case be rigorously extended to non-radial domains?
• RQ4 What role do the Green's function and regular part of the Green's function play in determining the extinction profile?
• RQ5 Under what geometric and algebraic conditions on the points $ q_1, \dots, q_k $ does a solution with multiple peaks exist?

Key findings

• The solution exhibits extinction rate $ \|w(\cdot,\tau)\|_{L^\infty(\Omega)} = \gamma_0 (T-\tau)^{\frac{n+2}{4}} |\ln(T-\tau)|^{\frac{n+2}{2(n-2)}} (1+o(1)) $ as $ \tau \to T^- $, confirming the logarithmic correction.
• The extinction profile is characterized by $ k $ localized peaks centered at fixed points $ q_j \in \Omega $, with amplitudes decaying as $ \tilde{\mu}_j(\tau) \sim \beta_j \left(\log \frac{T}{T-\tau}\right)^{-1/(n-2)} $.
• The matrix $ \mathcal{G}(q) $, involving the regular part of the Green's function and mutual Green's functions, must be positive definite for the solution to exist.
• The solution is asymptotically close to a sum of $ k $ singular profiles modeled on the standard solution $ U(y) = (1+|y|^2)^{-(n-2)/2} $, corrected by the regular part $ H(x,q_j) $.
• The outer error term $ \tilde{\varphi}(x,\tau) \to 0 $ uniformly away from the points $ q_j $, ensuring the profile is localized.
• The existence of such initial data $ w_0 $ is established via a fixed-point argument in a weighted function space, under the condition that the matrix $ \mathcal{G}(q) $ is positive definite.