Refined blowup analysis and nonexistence of Type II blowups for an energy critical nonlinear heat equation

2021/01/18 by Wang, Kelei, Wei, Juncheng · 1 citation
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2101.07186

Abstract

We consider the energy critical semilinear heat equation \\beginaligned amp;∂_t u-Δu =|u|^(4)/(n-2)u amp;in \mathbb Rⁿ×(0,T),
amp;u(x,0)=u₀(x), \endaligned. where n≥ 3, u₀∈ L^∞(\mathbb Rⁿ), and T∈ \mathbb R⁺ is the first blow up time. We prove that if n ≥ 7 and u₀ ≥ 0, then any blowup must be of Type I, i.e., ‖u(⋅, t)‖_L^∞(\mathbb Rⁿ)≤ C(T-t)^-(1)/(p-1). A similar result holds for bounded convex domains. The proof relies on a reverse inner-outer gluing mechanism and delicate analysis of bubbling behavior (bubbling tower/cluster).

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• Blow-up rate for the subcritical semilinear heat equation in non-convex domains

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