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

TL;DR

This paper studies the long-term oscillatory behavior of solutions to the 6D energy critical heat equation, revealing solutions that exhibit unbounded scaling parameters and differ from typical H^1 solutions.

Contribution

It constructs radially symmetric solutions with oscillating scale parameters, showing complex dynamics in the critical Sobolev space.

Findings

Existence of solutions with \\lambda(t) oscillating between 0 and infinity.

Solutions approximate a rescaled ground state with vanishing error.

Demonstrates different dynamics in \\dot H^1(\\mathbb{R}^6) compared to H^1(\\mathbb{R}^6).

Abstract

Long time dynamics of solutions to the 6D energy critical heat equation $u_t=\Delta u+|u|^{p-1}u$ 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) = \lambda(t)^{-\frac{n-2}{2}} {\sf Q}(\tfrac{x}{\lambda(t)}) + \text{error} (x,t), \end{align*} where the function \( \lambda(t) \) satisfies: \begin{itemize} \item $\dis\lim_{t\to\infty}\|\text{error}(\cdot,t)\|_{\dot H_x^1(\R^6)}=0$, \item $\dis\liminf_{t\to\infty}\lambda(t)=0$, \item $\dis\limsup_{t\to\infty}\lambda(t)=\infty$. \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 \( H^1(\mathbb{R}^n) \).