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

Tomoyuki Tanaka; Kyouhei Wakasa

doi:10.3934/dcds.2021048

Article Contents

2021, Volume 41, Issue 10: 4545-4566. Doi: 10.3934/dcds.2021048

This issue Previous Article Floquet solutions for the Schrödinger equation with fast-oscillating quasi-periodic potentials Next Article On conormal derivative problem for parabolic equations with Dini mean oscillation coefficients

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

Tomoyuki Tanaka^1, , and
Kyouhei Wakasa^2,

1.
Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan
2.
Department of Creative Engineering, National Institute of Technology, Kushiro College, 2-32-1 Otanoshike-Nishi, Kushiro-Shi, Hokkaido 084-0916, Japan

^* Corresponding author: Tomoyuki Tanaka
^* Corresponding author: Tomoyuki Tanaka

Received: August 31, 2020

Revised: December 31, 2020

Early access: March 2021

Published: October 2021

The first author is supported by JSPS KAKENHI Grant Number JP20J12750. The second author is supported by the Grant-in-Aid for Scientific Research (B) (No.18H01132), the Grant-in-Aid for Scientific Research (B) (No.19H01795) and Young Scientists Research (No. 20K14351), Japan Society for the Promotion of Science

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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 [13]). When $ 2< \gamma<3 $, we also prove unique global existence of solutions for small data.

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Mathematics Subject Classification: Primary: 35B33; Secondary: 35B44, 35L05, 35L71, 35B45.

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