Long-time dynamics for the energy critical heat equation in $R^5$

Abstract

We investigate the long-time behavior of global solutions to the energy critical heat equation in $R^5$ \begin{equation*} \begin{cases} \pp_t u=Δu+|u|^{\frac{4}{3}} u ~&\mbox{ in }~ R^5 \times (t_0,\infty), u(\cdot,t_0)=u_0~&\mbox{ in }~ R^5. \end{cases} \end{equation*} For $t_0$ sufficiently large, we show the existence of positive solutions for a class of initial value $u_0(x)\sim |x{|}^{-\gamma }$ as $|x|\to \infty$ with $\gamma >\frac{3}{2}$ such that the global solutions behave asymptotically \begin{equation*} \| u(\cdot,t) \|_{L^\infty (\R^5)} \sim \begin{cases} t^{-\frac{3(2-γ)}{2}} ~&\mbox{ if }~ \frac32<γ<2 (\ln t)^{-3} ~&\mbox{ if }~ γ=2 1 ~&\mbox{ if }~ γ>2 \end{cases} \mbox{ \ for \ } t >t_0, \end{equation*} which is slower than the self-similar time decay $t^{-\frac{3}{4}}$. These rates are inspired by Fila-King \cite[Conjecture 1.1]{FilaKing12}.

19 pages; comments welcome