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

TL;DR

This paper investigates the wave equation with a time-dependent damping and a cubic convolution, establishing the critical exponent for global existence versus blow-up, and analyzing the solution's asymptotic behavior in three dimensions.

Contribution

It identifies the critical exponent as 0, showing how scale-invariant damping shifts the threshold from 2 to 0, and provides results on global existence and lifespan estimates.

Findings

Critical exponent for global existence is 0.

Global solutions exist for supercritical case $\gamma ightarrow 0$.

Finite lifespan and blow-up occur in the subcritical case $\gamma<0$.

Abstract

In the present paper, we study the Cauchy problem for the wave equation with a time-dependent scale invariant damping $\frac{2}{1+t}\partial_t v$ and a cubic convolution $(|x|^{-\gamma}*v^2)v$ with $\gamma\in \left(-\frac{1}{2},3\right)$ in three spatial dimension for initial data $\left(v(x,0),\partial_tv(x,0)\right)\in C^2(\mathbb{R}^3)\times C^1(\mathbb{R}^3)$ with a compact support, where $v=v(x,t)$ is an unknown function to the problem on $\mathbb{R}^3\times[0,T)$. Here $T$ denotes a maximal existence time of $v$. The first aim of the present paper is to prove unique global existence of the solution to the problem and asymptotic behavior of the solution in the supercritical case $\gamma\in (0,3)$, and show a lower estimate of the lifespan in the critical or subcritical case $\gamma\in \left(-\frac{1}{2},0\right]$. The essential part for their proofs is to derive a weaker estimate under the weaker condition than the case without damping and to recover the weakness by the effect of the dissipative term. The second aim of the present paper is to prove a small data blow-up and the almost sharp upper estimate of the lifespan for positive data with a compact support in the subcritical case $\gamma\in \left(-\frac{1}{2},0\right)$. The essential part for the proof is to refine the argument for the proof of Theorem 6.1 in \cite{H20} to obtain the upper estimate of the lifespan. Our two results determine that a critical exponent $\gamma_c$ which divides global existence and blow-up for small solutions is $0$, namely $\gamma_c=0$. As the result, we can see that the critical exponent shift from $2$ to $0$ due to the effect of the scale invariant damping term.