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.$\frac{2}{1+t}{\partial }_{t}v$ and a cubic convolution $(|x{|}^{-\gamma }\ast v^2)v$ 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 $|x|\to \infty$. 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 ($n\ge 4$).

arXiv comment26 pages, no figures

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Cited by in corpus (1)

• Critical exponent for the wave equation with a time-dependent scale invariant damping and a cubic convolution