Long-time dynamics for the energy critical heat equation in R⁵

2023/08/18 by Zaizheng Li, Li, Zaizheng, Qidi Zhang +5
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2308.09754

openalex publication_date 2023/08/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We investigate the long-time behavior of global solutions to the energy critical heat equation in R⁵ \begincases \pp_t u=Δu+|u|^(4)/(3) u ~amp; in ~ R⁵ × (t₀,∞), u(⋅,t₀)=u₀~amp; in ~ R⁵. \endcases For t₀ sufficiently large, we show the existence of positive solutions for a class of initial value u₀(x)∼ |x|^-γ as |x|→ ∞ with γ>\frac32 such that the global solutions behave asymptotically ‖ u(⋅,t) ‖_{L^∞ (\R⁵)} ∼ \begincases t^{-(3(2-γ))/(2)} ~amp; if ~ \frac32lt;γlt;2 (ln t)^-3 ~amp; if ~ γ=2 1 ~amp; if ~ γgt;2 \endcases for t gt;t₀, which is slower than the self-similar time decay t^-(3)/(4). These rates are inspired by Fila-King \cite[Conjecture 1.1]FilaKing12.

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