On Fila-King Conjecture in Dimension Four

2022/10/09 by Juncheng Wei, Qidi Zhang, Wei, Juncheng +3
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2210.04352

openalex publication_date 2022/10/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the following Cauchy problem for the four-dimensional energy critical heat equation \begincases u_t=Δu+u³ ~amp; in ~ \mathbb R⁴ × (0,∞),
u(x,0)=u₀(x) ~amp; in ~ \mathbb R⁴. \endcases We construct a positive infinite time blow-up solution u(x,t) with the blow-up rate ‖ u(⋅,t)‖_L^∞(\mathbb R⁴) ∼ ln t as t→ ∞ and show the stability of the infinite time blow-up. This gives a rigorous proof of a conjecture by Fila and King \cite[Conjecture 1.1]filaking12.

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