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

TL;DR

This paper proves small data blow-up and lifespan estimates for a wave equation with time-dependent damping and cubic convolution in high dimensions, addressing slowly decaying initial data.

Contribution

It provides the first blow-up results for wave equations with cubic convolution in dimensions four and higher, including lifespan estimates for slowly decaying initial data.

Findings

Established small data blow-up for the wave equation with cubic convolution.

Derived upper lifespan estimates for solutions with slowly decaying initial data.

Extended blow-up analysis to high spatial dimensions (n ≥ 4).

Abstract

In the present paper, we study the Cauchy problem for the wave equation with a time-dependent scale invariant damping, i.e.$\frac{2}{1+t}\partial_t v$ and a cubic convolution $(|x|^{-\gamma}*v^2)v$ with $\gamma\in (0,n)$, where $v=v(x,t)$ is an unknown function on $\mathbb{R}^n\times[0,T)$. 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 $(v(x,0),\partial_t v(x,0))$ such as $\partial_t v(x,0)=O(|x|^{-(1+\nu)})$ as $|x|\rightarrow\infty$. Here $\nu$ belongs to the scaling supercritical case $\nu<\frac{n-\gamma}{2}$. Our main new contribution is to estimate the convolution term in high spatial dimensions, i.e. $n\ge 4$. This paper is the first blow-up result to treat wave equations with the cubic convolution in high spatial dimensions ($n\ge 4$).