Weakly non-radiative radial solutions to 3D energy subcritical wave equations

TL;DR

This paper investigates the asymptotic behavior of weakly non-radiative solutions to 3D energy subcritical wave equations, revealing new behaviors and establishing conditions under which solutions resemble elliptic solutions with specific decay.

Contribution

It demonstrates that radial weakly non-radiative solutions in the subcritical case can differ from known asymptotics and links solutions with initial data in the critical Sobolev space to elliptic solutions.

Findings

Radial solutions can have diverse asymptotics in the subcritical case.

Solutions with critical Sobolev initial data match elliptic solutions asymptotically.

Examples show deviations from classical asymptotic behavior in the subcritical regime.

Abstract

In this work we consider the energy subcritical 3D wave equation $\partial_t^2 u - \Delta u = \pm |u|^{p-1} u$ and discuss its (weakly) non-radiative solutions, i.e. the solutions defined in an exterior region $\{(x,t): |x|>|t|+R\}$ with $R\geq 0$ satisfying \[ \lim_{t\rightarrow \pm\infty} \int_{|x|>|t|+R} \left(|\nabla u(x,t)|^2 + |u_t(x,t)|^2\right) dx = 0. \] It has been known that any radial weakly non-radiative solution to the linear wave equation is a multiple of $1/|x|$. In addition, any radial weakly non-radiative solutions $u$ to the energy critical wave equation must possess a similar asymptotic behaviour, i.e. $u(x,t)\simeq C/|x|$ when $|x|$ is large. In this work we give examples to show that radial weakly non-radiative solutions to energy subcritical equation ($3<p<5$) may possess a much different asymptotic behaviour. However, a radial weakly non-radiative solution $u$ with initial data in the critical Sobolev space $\dot{H}^{s_p}\times \dot{H}^{s_p-1}(\mathbb{R}^3)$ must coincide with a $C^2$ solution $W$ to the elliptic equation $-\Delta W = -|W|^{p-1} W$ so that $u(x,t) \equiv W(x) \simeq C/|x|$ when $|x|$ is large.