The biharmonic nonlinear heat equation standing-ring conjecture

Consider the radially symmetric biharmonic nonlinear heat equation

$$u_t+\Delta^2u-|u|^{2\sigma}u=0,\qquad \sigma>0,$$

and let $L(t)$ and $r_{\max}(t)$ denote the ring similarity scale and radius. Let $v_B(t,x)$ be a singular peak-type solution of the one-dimensional biharmonic nonlinear heat equation. Biharmonic nonlinear heat standing-ring conjecture. Singular standing-ring solutions exist, and in the ring region $u\sim u_B$ for $r-r_{\max}=\mathcal{O}(L)$, where

$$u_B(t,r)=v_B\bigl(t,x=r-r_{\max}(t)\bigr).$$

The profile and blowup rate should therefore coincide with those of the one-dimensional peak solutions; no resolution evidence is supplied.