Blow-up for a fully fractional heat equation

2022/12/20 by Ferreira, Raúl, de Pablo, Arturo · 2 citations
#26A33 #35B44 #35R09 #35R11 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2212.10603

Abstract

We study the existence and behaviour of blowing-up solutions to the fully fractional heat equation M u=u^p, x∈ℝ^N, 00, where M is a nonlocal operator given by a space-time kernel M(x,t)=c_N,σt^{-\frac N2-1-σ}e^-(|x|2)/(4t)1_\t>0\, 0<σ<1. This operator coincides with the fractional power of the heat operator, M=(∂_t-Δ)^σ defined through semigroup theory. We characterize the global existence exponent p₀=1 and the Fujita exponent p_*=1+(2σ)/(N+2(1-σ)), and study the rate at which the blowing-up solutions below p_* tend to infinity, ‖u(⋅,t)‖_∞∼ (T-t)^-\fracσp-1.

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