Small data blow-up for the wave equation with a time-dependent scale invariant damping and a cubic convolution for slowly decaying initial data

2020/01/22 by Masahiro Ikeda, Tomoyuki Tanaka, Ikeda, Masahiro +3
Mathematics · #35B44 #35L05 #35L71 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.2001.07985

openalex publication_date 2020/01/22 · openalex created_date 2020/01/30 · openalex updated_date 2026/07/28

Abstract

In the present paper, we study the Cauchy problem for the wave equation with a time-dependent scale invariant damping, i.e.(2)/(1+t)∂_t v and a cubic convolution (|x|^-γ*v²)v with γ∈ (0,n), where v=v(x,t) is an unknown function on ℝⁿ×[0,T). Our aim of the present paper is to prove a small data blow-up result and show an upper estimate of lifespan of the problem for slowly decaying positive initial data (v(x,0),∂_t v(x,0)) such as ∂_t v(x,0)=O(|x|^-(1+ν)) as |x|→∞. Here ν belongs to the scaling supercritical case ν

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