From $p_0(n)$ to $p_0(n+2)$

TL;DR

This paper investigates the critical exponent for small data solutions to a semi-linear wave equation with a damping term, establishing blow-up results in the subcritical range and existence results above it, leading to a conjecture relating to the Strauss exponent.

Contribution

It identifies the critical exponent for the damped wave equation and conjectures its relation to the Strauss exponent for the classical wave equation.

Findings

Blow-up in finite time for p in (1, p_2(n)]

Existence of solutions for p > p_2(n) in dimensions 2 and 3

Conjecture p_2(n) = p_0(n+2) for n ≥ 2

Abstract

In this note we study the global existence of small data solutions to the Cauchy problem for the semi-linear wave equation with a not effective scale-invariant damping term, namely \[ v_{tt}-\triangle v + \frac2{1+t}\,v_t = |v|^p, \qquad v(0,x)=v_0(x),\quad v_t(0,x)=v_1(x), \] where $p>1$, $n\ge 2$. We prove blow-up in finite time in the subcritical range $p\in(1,p_2(n)]$ and an existence result for $p>p_2(n)$, $n=2,3$. In this way we find the critical exponent for small data solutions to this problem. All these considerations lead to the conjecture $p_2(n)=p_0(n+2)$ for $n\ge2$, where $p_0(n)$ is the Strauss exponent for the classical wave equation.