On the critical decay for the wave equation with a cubic convolution in 3D

TL;DR

This paper investigates the wave equation with a cubic convolution in 3D, establishing lifespan estimates at the critical decay rate and proving global existence for small data when the convolution parameter exceeds 2.

Contribution

It provides optimal lifespan estimates for the critical case and confirms global existence results for the supercritical case, addressing conjectures in the field.

Findings

Optimal lifespan estimate for b3=2

Global existence for small data when b3>2

Affirmative answer to Kubo conjecture

Abstract

We consider the wave equation with a cubic convolution $\partial_t^2 u-\Delta u=(|x|^{-\gamma}*u^2)u$ in three space dimensions. Here, $0<\gamma<3$ and $*$ stands for the convolution in the space variables. It is well known that if initial data are smooth, small and compactly supported, then $\gamma\ge2$ assures unique global existence of solutions. On the other hand, it is also well known that solutions blow up in finite time for initial data whose decay rate is not rapid enough even when $2\le \gamma<3$. In this paper, we consider the Cauchy problem for $2\le \gamma<3$ in the space-time weighted $L^\infty$ space in which functions have critical decay rate. When $\gamma=2$, we give an optimal estimate of the lifespan. This gives an affirmative answer to the Kubo conjecture (see Remark right after Theorem 2.1 in Kubo(2004)). When $2<\gamma<3$, we also prove unique global existence of solutions for small data.