A slow blow up solution for the four dimensional energy critical semi linear heat equation

2022/04/24 by Tongtong Li, Liming Sun, Li, Tongtong +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.2204.11201

openalex publication_date 2022/04/24 · openalex created_date 2022/04/28 · openalex updated_date 2026/07/28

Abstract

We consider the energy critical four dimensional semi-linear heat equation ∂_tv-Δv-v³=0, (t,x)∈ ℝ× ℝ⁴. Formal computation of Filippas et al. (R. Soc. Lond. Proc. 2000) conjectures the existence of a sequence of type II blow-up solutions with various blow-up rates ‖v(t)‖_{L^∞(ℝ⁴)}≈ \frac|log(T-t)|^(2L)/(2L-1)(T-t)^L , L=1,2,⋯. Schweyer (J. Funct. Anal. 2012) rigorously constructs a type II blow-up solution for the case L=1. In this paper, we show the existence of type II blow-up solution for L=2. The method here could be generalized to deal with all the cases L≥ 2.

Related

• A higher speed type II blowup for the five dimensional energy critical heat equation
• Type II Blow Up for the Four Dimensional Energy Critical Semi Linear Heat Equation
• Refined blowup analysis and nonexistence of Type II blowups for an energy critical nonlinear heat equation
• Type II Finite time blow-up for the three dimensional energy critical heat equation
• A higher speed type II blow-up for the 4-d Fujita equation