Oscillatory behavior of solutions to the critical Fujita equation in 6D

Long time dynamics of solutions to the 6D energy critical heat equation $u_t=\mathrm{\Delta }u+\mathrm{\mid }u{\mathrm{\mid }}^{p-1}u$ on ${\mathbb{R}}^6\times (0,\mathrm{\infty })$ is investigated. It is shown that there exists a radially symmetric global solution $u(x,t)\in C([0,\mathrm{\infty });{\dot{H}}^1({\mathbb{R}}^6))$ of the form \begin{align*} u(x,t) = λ(t)^{-\frac{n-2}{2}} {\sf Q}(\tfrac{x}{λ(t)}) + $\text{error}$ (x,t), \end{align*} where the function $\lambda (t)$ satisfies: \begin{itemize} \item $\text{dis}{\lim }_{t\to \mathrm{\infty }}\mathrm{\parallel }\text{error}(\cdot ,t){\mathrm{\parallel }}_{{\dot{H}}_x^1({\mathbb{R}}^6)}=0$, \item $\text{dis}{\mathrm{lim\hspace{0.17em}inf}}_{t\to \mathrm{\infty }}\lambda (t)=0$, \item $\text{dis}{\mathrm{lim\hspace{0.17em}sup}}_{t\to \mathrm{\infty }}\lambda (t)=\mathrm{\infty }$. \end{itemize} The solutions constructed here demonstrate that the dynamical behavior in ${\dot{H}}^1({\mathbb{R}}^n)$ can differ significantly from the behavior in $H^1({\mathbb{R}}^n)$.