Late-time decay-rate conjecture for nonlinear wave modes

Consider the nonlinear wave equation

$$-\partial_t^2\Phi+\Delta\Phi=\mu\Phi|\Phi|^{p-1},\qquad \Phi:\mathbb{R}\times\mathbb{R}^3\to\mathbb{R},$$

with $p\in\mathbb{N}$ and $p\geqslant3$. Expand the field in spherical harmonics,

$$\Phi(t,r,\theta,\varphi)=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}\Phi_{lm}(t,r)Y_{lm}(\theta,\varphi).$$

At fixed finite radius $r$, the modes are expected to decay asymptotically as

$$\Phi_{lm}(t,r)\sim t^{-q_l},\qquad q_l=\max(l+p-1,2l+2).$$

On slices $\Sigma_{\tilde t}$ approaching future null infinity, where $t,r\to\infty$, $t-r\to\tilde t$, and the conformal factor satisfies $\Omega(r)=\operatorname{O}(r^{-1})$, define the conformally rescaled field $\tilde\Phi=\Omega(r)^{(1-n)/2}\Phi$. Late-time decay-rate conjecture. The mode $\Phi_{lm}$ decays at fixed finite radius according to the exponent $q_l$ above, while at future null infinity the rescaled modes satisfy

$$\tilde\Phi_{lm}(\tilde t)\sim\tilde t^{-\tilde q_l},\qquad \tilde q_l=\max(p-2,l+1).$$

The claim summarizes numerically observed nonlinear tails for both focusing and defocusing nonlinearities and for nonspherically symmetric evolutions. The finite-radius exponents are expected to be independent of the azimuthal index $m$ and of the extraction radius, whereas future null infinity has different, generally smaller, decay rates. A general analytical justification of these asymptotics remains open.