Refined asymptotics for the Cauchy problem for the fast $p$-Laplace evolution equation

TL;DR

This work analyzes the long-time behavior of nonnegative solutions to the fast p-Laplace evolution equation on ${\mathbb{R}^N}$ for $1<p<2$, revealing a detailed dichotomy between the good fast diffusion range ${p_c}<p<2$ (mass-conserving) and the very fast diffusion range $1<p\le p_c$ (extinction-prone). It develops a refined entropy method to obtain quantitative convergence to Barenblatt-type profiles, including explicit rates in relative error and, for radial data, convergence of gradients; it also identifies new critical exponents, such as ${p_M}$ and ${p_D}$, that govern the decay and regularity properties across dimensions. The authors translate results between original and self-similar variables, extend the framework to include extinction-profile analysis via pseudo-Barenblatt solutions, and establish a rigorous foundation for the basin of attraction of Barenblatt profiles, highlighting both sharp results and open questions for nonradial data and near-threshold exponents. The second half of the paper develops derivative-regularity results for radial solutions using a radial-derivative correspondence with weighted nonlinear diffusion, yielding exponential control of radial derivatives under suitable hypotheses. Overall, the work provides a comprehensive, entropy-driven, quantitative description of asymptotics in both diffusion regimes, with significant implications for the understanding of nonlinear diffusion dynamics and potential extensions to more general anisotropic or doubly nonlinear settings.

Abstract

Our focus is on the fast diffusion equation driven by the $p$-Laplacian operator, that is $\partial_t u=Δ_p u$ with $1<p<2$, posed in the whole space $\mathbb{R}^N$, $N\geq 2$. The nonnegative solutions are expected to converge in time toward a stationary profile. While such convergence had been previously established for $p$ close to $2$, no quantitative rates were known, and the asymptotic behaviour remained poorly understood across the full fast diffusion range. In fact, the long time behaviour of solutions to the $p$-Laplace Cauchy problem drastically change in different subranges of the $p$. Some of them are analysed here for the first time. In this work, we provide the convergence rates for nonnegative, integrable solutions in the so-called good fast diffusion range, $p_c=\tfrac{2N}{N+1} <p<2$, where mass is conserved. We prove that solutions converge to a self-similar profile with matching mass, with explicit rates measured in relative error. Our constructive proof is based on a new entropy method that remains effective even when the entropy is not displacement convex -- where optimal transport techniques fail. In the very fast diffusion range $1<p<p_c$, we give the first asymptotic analysis near the extinction time. We uncover new critical exponents -- especially in high dimensions -- that give rise to markedly different qualitative behaviour depending on the value of $p$. We also establish convergence rates for the gradients of radial solutions in the good fast diffusion range, again measured in relative error. Finally, we analyze the structural properties required for the entropy method to apply, thereby opening a broader investigation into the basin of attraction of Barenblatt-type profiles, particularly in the singular case of $p$ close to $1$.

Figures (1)

• Figure 1: Special values of parameter $p$ when $N\geq 7$. For $p>p_D$ the entropy is displacement convex, for $p>p_M$ Barenblatts have finite weighted ${p'}$-moments, for $p>p_c$ Barenblatt solutions are integrable and the mass is conserved, for $p\in(1,p_1)\cup(p_2,2)$ difference of two Barenblatt solutions is integrable, $p_Y$ is a gradient regularity threshold, for $p=\frac{3}{2}$ we have $\gamma=0$ and the entropy functional is not defined.

Theorems & Definitions (29)

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• proof : Proof of Theorem \ref{['theo:conv-L1-RE']}
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