Oscillatory behavior of solutions to the critical Fujita equation in 6D
Junichi Harada
TL;DR
The article analyzes the long-time dynamics of the energy-critical Fujita equation in 6 dimensions, constructing three classes of global radially symmetric solutions with a leading Aubin–Talenti profile $u(x,t)\approx\lambda(t)^{-(n-2)/2}Q(x/\lambda(t))$ and a vanishing error in $\dot H^1$. It uses an inner-outer gluing framework and matched asymptotics to manipulate a modulation parameter $\lambda(t)$, producing both monotone and oscillatory behaviors; in particular, the oscillatory solution exhibits $\liminf_{t\to\infty}\lambda(t)=0$ and $\limsup_{t\to\infty}\lambda(t)=\infty$. The construction relies on a careful analysis of the linearized operator around the ground state, tail decay via the linear heat equation, and a fixed-point argument that couples the inner core with an outer tail. This work highlights a novel dynamic regime in which $\dot H^1$ dynamics can diverge significantly from $H^1$ behavior, revealing intricate, nonstatic evolution near the energy-critical ground state in the 6D setting.
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 \( H^1(\mathbb{R}^n) \).