Oscillatory behavior of solutions to the critical Fujita equation in 6D (2511.17891v1)

Abstract: Long time dynamics of solutions to the 6D energy critical heat equation $u_t=Δu+|u|^{p-1}u$ on $\R^6\times(0,\infty)$ is investigated. It is shown that there exists a radially symmetric global solution $u(x,t)\in C([0,\infty);\dot H^1(\R^6))$ 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 ( λ(t) ) satisfies: \begin{itemize} \item $\dis\lim_{t\to\infty}|\text{error}(\cdot,t)|{\dot H_x^1(\R^6)}=0$, \item $\dis\liminf{t\to\infty}λ(t)=0$, \item $\dis\limsup_{t\to\infty}λ(t)=\infty$. \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) ).