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

In this work we consider the energy subcritical 3D wave equation $\partial_t^2 u - Δu = \pm |u|^{p-1} u$ 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 $-ΔW = -|W|^{p-1} W$ so that $u(x,t) \equiv W(x) \simeq C/|x|$ when $|x|$ is large.