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

Abstract

In the present paper, we study the Cauchy problem for the wave equation with a time-dependent scale invariant damping, i.e. 21+t∂tv 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),∂tv(x,0)) such as ∂tv(x,0)=O(|x|−(1+ν)) as |x|→∞. Here ν belongs to the scaling supercritical case ν<n−γ2. The proof of our main result is based on the combination of the arguments in the papers Takamura et al. (2010) and Takamura (1995). Especially, our main new contribution is to estimate the convolution term in high spatial dimensions, i.e. n≥4. This paper is the first blow-up result to treat wave equations with the cubic convolution in high spatial dimensions (n≥4).